Finite intersection propertyFrom Wikipedia, the free encyclopedia

Contents

1 Finite intersection property 11.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.6 Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 General topology 32.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 A topology on a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2.1 Basis for a topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2.2 Subspace and quotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2.3 Examples of topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 Continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3.1 Alternative denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3.3 Homeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3.4 Dening topologies via continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.4 Compact sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.5 Connected sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.5.1 Connected components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5.2 Disconnected spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5.3 Path-connected sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.6 Products of spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.7 Separation axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.8 Countability axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.9 Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.10 Baire category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.11 Main areas of research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.11.1 Continuum theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.11.2 Pointless topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

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ii CONTENTS

2.11.3 Dimension theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.11.4 Topological algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.11.5 Metrizability theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.11.6 Set-theoretic topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.12 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.13 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.14 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Nested intervals 163.1 Higher dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 Tychonos theorem 184.1 Topological denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.3 Proofs of Tychonos theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.4 Tychonos theorem and the axiom of choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.5 Proof of the axiom of choice from Tychonos theorem . . . . . . . . . . . . . . . . . . . . . . . 204.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5 Uncountable set 225.1 Characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.4 Without the axiom of choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.8 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5.8.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.8.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.8.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Chapter 1

Finite intersection property

In general topology, a branch of mathematics, a collectionA of subsets of a setX is said to have the nite intersectionproperty (FIP) if the intersection over any nite subcollection of A is nonempty. It has the strong nite intersectionproperty (SFIP) if the intersection over any nite subcollection of A is innite.A centered system of sets is a collection of sets with the nite intersection property.

1.1 DenitionLetX be a set withA = fAigi2I a family of subsets ofX . Then the collectionA has the nite intersection property(FIP), if any nite subcollection J I has non-empty intersectionTi2J Ai:1.2 DiscussionClearly the empty set cannot belong to any collection with the nite intersection property. The condition is triviallysatised if the intersection over the entire collection is nonempty (in particular, if the collection itself is empty), andit is also trivially satised if the collection is nested, meaning that the collection is totally ordered by inclusion (equiv-alently, for any nite subcollection, a particular element of the subcollection is contained in all the other elements ofthe subcollection), e.g. the nested sequence of intervals (0, 1/n). These are not the only possibilities however. Forexample, if X = (0, 1) and for each positive integer i, Xi is the set of elements of X having a decimal expansion withdigit 0 in the i'th decimal place, then any nite intersection is nonempty (just take 0 in those nitely many places and1 in the rest), but the intersection of all Xi for i 1 is empty, since no element of (0, 1) has all zero digits.The nite intersection property is useful in formulating an alternative denition of compactness: a space is compact ifand only if every collection of closed sets satisfying the nite intersection property has nonempty intersection itself.[1]This formulation of compactness is used in some proofs of Tychonos theorem and the uncountability of the realnumbers (see next section)

1.3 ApplicationsTheorem. Let X be a non-empty compact Hausdor space that satises the property that no one-point set is open.Then X is uncountable.Proof. We will show that if U X is nonempty and open, and if x is a point of X, then there is a neighbourhoodV U whose closure doesnt contain x (x may or may not be in U). Choose y in U dierent from x (if x is in U,then there must exist such a y for otherwise U would be an open one point set; if x isnt in U, this is possible sinceU is nonempty). Then by the Hausdor condition, choose disjoint neighbourhoodsW and K of x and y respectively.Then K U will be a neighbourhood of y contained in U whose closure doesnt contain x as desired.

Now suppose f : N X is a bijection, and let {xi : i N} denote the image of f. Let X be the rst open set

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2 CHAPTER 1. FINITE INTERSECTION PROPERTY

and choose a neighbourhood U1 X whose closure doesnt contain x1. Secondly, choose a neighbourhood U2 U1 whose closure doesnt contain x2. Continue this process whereby choosing a neighbourhood Un Un whoseclosure doesnt contain xn. Then the collection {Ui : i N} satises the nite intersection property and hence theintersection of their closures is nonempty (by the compactness of X). Therefore there is a point x in this intersection.No xi can belong to this intersection because xi doesnt belong to the closure of Ui. This means that x is not equal toxi for all i and f is not surjective; a contradiction. Therefore, X is uncountable.All the conditions in the statement of the theorem are necessary:1. We cannot eliminate the Hausdor condition; a countable set with the indiscrete topology is compact, has morethan one point, and satises the property that no one point sets are open, but is not uncountable.2. We cannot eliminate the compactness condition as the set of all rational numbers shows.3. We cannot eliminate the condition that one point sets cannot be open as a nite space given the discrete topologyshows.Corollary. Every closed interval [a, b] with a < b is uncountable. Therefore, R is uncountable.Corollary. Every perfect, locally compact Hausdor space is uncountable.Proof. Let X be a perfect, compact, Hausdor space, then the theorem immediately implies that X is uncountable.If X is a perfect, locally compact Hausdor space which is not compact, then the one-point compactication of X isa perfect, compact Hausdor space. Therefore the one point compactication of X is uncountable. Since removinga point from an uncountable set still leaves an uncountable set, X is uncountable as well.

1.4 ExamplesA lter has the nite intersection property by denition.

1.5 TheoremsLet X be nonempty, F 2X, F having the nite intersection property. Then there exists an F ultralter (in 2X) suchthat F F.See details and proof in Csirmaz & Hajnal (1994).[2] This result is known as ultralter lemma.

1.6 VariantsA family of sets A has the strong nite intersection property (sp), if every nite subfamily of A has inniteintersection.

1.7 References[1] A space is compact i any family of closed sets having p has non-empty intersection at PlanetMath.org.

[2] Csirmaz, Lszl; Hajnal, Andrs (1994), Matematikai logika (IN HUNGARIAN), Budapest: Etvs Lornd University.

Finite intersection property at PlanetMath.org.

Chapter 2

General topology

The Topologists sine curve, a useful example in point-set topology. It is connected but not path-connected.

In mathematics, general topology is the branch of topology that deals with the basic set-theoretic denitions andconstructions used in topology. It is the foundation ofmost other branches of topology, including dierential topology,geometric topology, and algebraic topology. Another name for general topology is point-set topology.The fundamental concepts in point-set topology are continuity, compactness, and connectedness:

Continuous functions, intuitively, take nearby points to nearby points. Compact sets are those that can be covered by nitely many sets of arbitrarily small size. Connected sets are sets that cannot be divided into two pieces that are far apart.

The words 'nearby', 'arbitrarily small', and 'far apart' can all be made precise by using open sets, as described below.If we change the denition of 'open set', we change what continuous functions, compact sets, and connected sets are.Each choice of denition for 'open set' is called a topology. A set with a topology is called a topological space.

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4 CHAPTER 2. GENERAL TOPOLOGY

Metric spaces are an important class of topological spaces where distances can be assigned a number called a metric.Having a metric simplies many proofs, and many of the most common topological spaces are metric spaces.

2.1 HistoryGeneral topology grew out of a number of areas, most importantly the following:

the detailed study of subsets of the real line (once known as the topology of point sets, this usage is now obsolete) the introduction of the manifold concept the study of metric spaces, especially normed linear spaces, in the early days of functional analysis.

General topology assumed its present form around 1940. It captures, one might say, almost everything in the intuitionof continuity, in a technically adequate form that can be applied in any area of mathematics.

2.2 A topology on a setMain article: Topological space

Let X be a set and let be a family of subsets of X. Then is called a topology on X if:[1][2]

1. Both the empty set and X are elements of 2. Any union of elements of is an element of 3. Any intersection of nitely many elements of is an element of

If is a topology on X, then the pair (X, ) is called a topological space. The notation Xmay be used to denote a setX endowed with the particular topology .The members of are called open sets in X. A subset of X is said to be closed if its complement is in (i.e., itscomplement is open). A subset of X may be open, closed, both (clopen set), or neither. The empty set and X itselfare always both closed and open.

2.2.1 Basis for a topologyMain article: Basis (topology)

A base (or basis) B for a topological space X with topology T is a collection of open sets in T such that every open setin T can be written as a union of elements of B.[3][4] We say that the base generates the topology T. Bases are usefulbecause many properties of topologies can be reduced to statements about a base that generates that topologyandbecause many topologies are most easily dened in terms of a base that generates them.

2.2.2 Subspace and quotientEvery subset of a topological space can be given the subspace topology in which the open sets are the intersectionsof the open sets of the larger space with the subset. For any indexed family of topological spaces, the product can begiven the product topology, which is generated by the inverse images of open sets of the factors under the projectionmappings. For example, in nite products, a basis for the product topology consists of all products of open sets. Forinnite products, there is the additional requirement that in a basic open set, all but nitely many of its projectionsare the entire space.A quotient space is dened as follows: if X is a topological space and Y is a set, and if f : X Y is a surjectivefunction, then the quotient topology on Y is the collection of subsets of Y that have open inverse images under f. In

2.3. CONTINUOUS FUNCTIONS 5

other words, the quotient topology is the nest topology on Y for which f is continuous. A common example of aquotient topology is when an equivalence relation is dened on the topological space X. The map f is then the naturalprojection onto the set of equivalence classes.

2.2.3 Examples of topological spaces

A given set may have many dierent topologies. If a set is given a dierent topology, it is viewed as a dierenttopological space. Any set can be given the discrete topology in which every subset is open. The only convergentsequences or nets in this topology are those that are eventually constant. Also, any set can be given the trivial topology(also called the indiscrete topology), in which only the empty set and the whole space are open. Every sequence andnet in this topology converges to every point of the space. This example shows that in general topological spaces,limits of sequences need not be unique. However, often topological spaces must be Hausdor spaces where limitpoints are unique.There are many ways to dene a topology on R, the set of real numbers. The standard topology on R is generatedby the open intervals. The set of all open intervals forms a base or basis for the topology, meaning that every openset is a union of some collection of sets from the base. In particular, this means that a set is open if there exists anopen interval of non zero radius about every point in the set. More generally, the Euclidean spaces Rn can be givena topology. In the usual topology on Rn the basic open sets are the open balls. Similarly, C, the set of complexnumbers, and Cn have a standard topology in which the basic open sets are open balls.Every metric space can be given a metric topology, in which the basic open sets are open balls dened by the metric.This is the standard topology on any normed vector space. On a nite-dimensional vector space this topology is thesame for all norms.Many sets of linear operators in functional analysis are endowed with topologies that are dened by specifying whena particular sequence of functions converges to the zero function.Any local eld has a topology native to it, and this can be extended to vector spaces over that eld.Every manifold has a natural topology since it is locally Euclidean. Similarly, every simplex and every simplicialcomplex inherits a natural topology from Rn.The Zariski topology is dened algebraically on the spectrum of a ring or an algebraic variety. On Rn or Cn, theclosed sets of the Zariski topology are the solution sets of systems of polynomial equations.A linear graph has a natural topology that generalises many of the geometric aspects of graphs with vertices andedges.The Sierpiski space is the simplest non-discrete topological space. It has important relations to the theory of com-putation and semantics.There exist numerous topologies on any given nite set. Such spaces are called nite topological spaces. Finite spacesare sometimes used to provide examples or counterexamples to conjectures about topological spaces in general.Any set can be given the conite topology in which the open sets are the empty set and the sets whose complementis nite. This is the smallest T1 topology on any innite set.Any set can be given the cocountable topology, in which a set is dened as open if it is either empty or its complementis countable. When the set is uncountable, this topology serves as a counterexample in many situations.The real line can also be given the lower limit topology. Here, the basic open sets are the half open intervals [a, b).This topology on R is strictly ner than the Euclidean topology dened above; a sequence converges to a point in thistopology if and only if it converges from above in the Euclidean topology. This example shows that a set may havemany distinct topologies dened on it.If is an ordinal number, then the set = [0, ) may be endowed with the order topology generated by the intervals(a, b), [0, b) and (a, ) where a and b are elements of .

2.3 Continuous functionsMain article: Continuous function

6 CHAPTER 2. GENERAL TOPOLOGY

Continuity is expressed in terms of neighborhoods: f is continuous at some point x X if and only if for any neigh-borhood V of f(x), there is a neighborhood U of x such that f(U) V. Intuitively, continuity means no matter howsmall V becomes, there is always a U containing x that maps inside V and whose image under f contains f(x).This is equivalent to the condition that the preimages of the open (closed) sets in Y are open (closed) in X. In metricspaces, this denition is equivalent to the -denition that is often used in analysis.An extreme example: if a set X is given the discrete topology, all functions

f : X ! T

to any topological space T are continuous. On the other hand, if X is equipped with the indiscrete topology and thespace T set is at least T0, then the only continuous functions are the constant functions. Conversely, any functionwhose range is indiscrete is continuous.

2.3.1 Alternative denitions

Several equivalent denitions for a topological structure exist and thus there are several equivalent ways to dene acontinuous function.

Neighborhood denition

Denitions based on preimages are often dicult to use directly. The following criterion expresses continuity interms of neighborhoods: f is continuous at some point x X if and only if for any neighborhood V of f(x), there isa neighborhood U of x such that f(U) V. Intuitively, continuity means no matter how small V becomes, there isalways a U containing x that maps inside V.If X and Y are metric spaces, it is equivalent to consider the neighborhood system of open balls centered at x and f(x)instead of all neighborhoods. This gives back the above - denition of continuity in the context of metric spaces.However, in general topological spaces, there is no notion of nearness or distance.Note, however, that if the target space is Hausdor, it is still true that f is continuous at a if and only if the limit off as x approaches a is f(a). At an isolated point, every function is continuous.

Sequences and nets

In several contexts, the topology of a space is conveniently specied in terms of limit points. In many instances, thisis accomplished by specifying when a point is the limit of a sequence, but for some spaces that are too large in somesense, one species also when a point is the limit of more general sets of points indexed by a directed set, knownas nets.[5] A function is continuous only if it takes limits of sequences to limits of sequences. In the former case,preservation of limits is also sucient; in the latter, a function may preserve all limits of sequences yet still fail to becontinuous, and preservation of nets is a necessary and sucient condition.In detail, a function f: X Y is sequentially continuous if whenever a sequence (xn) in X converges to a limit x,the sequence (f(xn)) converges to f(x).[6] Thus sequentially continuous functions preserve sequential limits. Everycontinuous function is sequentially continuous. If X is a rst-countable space and countable choice holds, then theconverse also holds: any function preserving sequential limits is continuous. In particular, if X is a metric space,sequential continuity and continuity are equivalent. For non rst-countable spaces, sequential continuity might bestrictly weaker than continuity. (The spaces for which the two properties are equivalent are called sequential spaces.)This motivates the consideration of nets instead of sequences in general topological spaces. Continuous functionspreserve limits of nets, and in fact this property characterizes continuous functions.

Closure operator denition

Instead of specifying the open subsets of a topological space, the topology can also be determined by a closure operator(denoted cl), which assigns to any subset A X its closure, or an interior operator (denoted int), which assigns to anysubset A of X its interior. In these terms, a function

2.3. CONTINUOUS FUNCTIONS 7

f : (X; cl)! (X 0; cl0)between topological spaces is continuous in the sense above if and only if for all subsets A of X

f(cl(A)) cl0(f(A)):That is to say, given any element x of X that is in the closure of any subset A, f(x) belongs to the closure of f(A).This is equivalent to the requirement that for all subsets A' of X'

f1(cl0(A0)) cl(f1(A0)):Moreover,

f : (X; int)! (X 0; int0)is continuous if and only if

f1(int0(A)) int(f1(A))for any subset A of X.

2.3.2 PropertiesIf f: X Y and g: Y Z are continuous, then so is the composition g f: X Z. If f: X Y is continuous and

X is compact, then f(X) is compact. X is connected, then f(X) is connected. X is path-connected, then f(X) is path-connected. X is Lindelf, then f(X) is Lindelf. X is separable, then f(X) is separable.

The possible topologies on a xed setX are partially ordered: a topology 1 is said to be coarser than another topology2 (notation: 1 2) if every open subset with respect to 1 is also open with respect to 2. Then, the identity map

idX: (X, 2) (X, 1)

is continuous if and only if 1 2 (see also comparison of topologies). More generally, a continuous function

(X; X)! (Y; Y )stays continuous if the topology Y is replaced by a coarser topology and/or X is replaced by a ner topology.

2.3.3 HomeomorphismsSymmetric to the concept of a continuous map is an open map, for which images of open sets are open. In fact, if anopen map f has an inverse function, that inverse is continuous, and if a continuous map g has an inverse, that inverseis open. Given a bijective function f between two topological spaces, the inverse function f1 need not be continuous.A bijective continuous function with continuous inverse function is called a homeomorphism.If a continuous bijection has as its domain a compact space and its codomain is Hausdor, then it is a homeomorphism.

8 CHAPTER 2. GENERAL TOPOLOGY

2.3.4 Dening topologies via continuous functionsGiven a function

f : X ! S;where X is a topological space and S is a set (without a specied topology), the nal topology on S is dened by lettingthe open sets of S be those subsets A of S for which f1(A) is open in X. If S has an existing topology, f is continuouswith respect to this topology if and only if the existing topology is coarser than the nal topology on S. Thus the naltopology can be characterized as the nest topology on S that makes f continuous. If f is surjective, this topology iscanonically identied with the quotient topology under the equivalence relation dened by f.Dually, for a function f from a set S to a topological space, the initial topology on S has as open subsets A of S thosesubsets for which f(A) is open in X. If S has an existing topology, f is continuous with respect to this topology if andonly if the existing topology is ner than the initial topology on S. Thus the initial topology can be characterized asthe coarsest topology on S that makes f continuous. If f is injective, this topology is canonically identied with thesubspace topology of S, viewed as a subset of X.More generally, given a set S, specifying the set of continuous functions

S ! Xinto all topological spaces X denes a topology. Dually, a similar idea can be applied to maps

X ! S:This is an instance of a universal property.

2.4 Compact setsMain article: Compact (mathematics)

Formally, a topological space X is called compact if each of its open covers has a nite subcover. Otherwise it iscalled non-compact. Explicitly, this means that for every arbitrary collection

fUg2Aof open subsets of X such that

X =[2A

U;

there is a nite subset J of A such that

X =[i2J

Ui:

Some branches of mathematics such as algebraic geometry, typically inuenced by the French school of Bourbaki,use the term quasi-compact for the general notion, and reserve the term compact for topological spaces that are bothHausdor and quasi-compact. A compact set is sometimes referred to as a compactum, plural compacta.Every closed interval in R of nite length is compact. More is true: In Rn, a set is compact if and only if it is closedand bounded. (See HeineBorel theorem).

2.5. CONNECTED SETS 9

Every continuous image of a compact space is compact.A compact subset of a Hausdor space is closed.Every continuous bijection from a compact space to a Hausdor space is necessarily a homeomorphism.Every sequence of points in a compact metric space has a convergent subsequence.Every compact nite-dimensional manifold can be embedded in some Euclidean space Rn.

2.5 Connected setsMain article: connected space

A topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. Otherwise, Xis said to be connected. A subset of a topological space is said to be connected if it is connected under its subspacetopology. Some authors exclude the empty set (with its unique topology) as a connected space, but this article doesnot follow that practice.For a topological space X the following conditions are equivalent:

1. X is connected.

2. X cannot be divided into two disjoint nonempty closed sets.

3. The only subsets of X that are both open and closed (clopen sets) are X and the empty set.

4. The only subsets of X with empty boundary are X and the empty set.

5. X cannot be written as the union of two nonempty separated sets.

6. The only continuous functions from X to {0,1}, the two-point space endowed with the discrete topology, areconstant.

Every interval in R is connected.The continuous image of a connected space is connected.

2.5.1 Connected componentsThe maximal connected subsets (ordered by inclusion) of a nonempty topological space are called the connectedcomponents of the space. The components of any topological space X form a partition of X: they are disjoint,nonempty, and their union is the whole space. Every component is a closed subset of the original space. It followsthat, in the case where their number is nite, each component is also an open subset. However, if their number isinnite, this might not be the case; for instance, the connected components of the set of the rational numbers are theone-point sets, which are not open.Let x be the connected component of x in a topological space X, and 0x be the intersection of all open-closed setscontaining x (called quasi-component of x.) Then x 0x where the equality holds if X is compact Hausdor orlocally connected.

2.5.2 Disconnected spacesA space in which all components are one-point sets is called totally disconnected. Related to this property, a space Xis called totally separated if, for any two distinct elements x and y of X, there exist disjoint open neighborhoods Uof x and V of y such that X is the union of U and V. Clearly any totally separated space is totally disconnected, butthe converse does not hold. For example take two copies of the rational numbers Q, and identify them at every pointexcept zero. The resulting space, with the quotient topology, is totally disconnected. However, by considering thetwo copies of zero, one sees that the space is not totally separated. In fact, it is not even Hausdor, and the conditionof being totally separated is strictly stronger than the condition of being Hausdor.

10 CHAPTER 2. GENERAL TOPOLOGY

2.5.3 Path-connected sets

This subspace of R is path-connected, because a path can be drawn between any two points in the space.

A path from a point x to a point y in a topological space X is a continuous function f from the unit interval [0,1] toX with f(0) = x and f(1) = y. A path-component of X is an equivalence class of X under the equivalence relation,which makes x equivalent to y if there is a path from x to y. The space X is said to be path-connected (or pathwiseconnected or 0-connected) if there is at most one path-component, i.e. if there is a path joining any two points inX. Again, many authors exclude the empty space.Every path-connected space is connected. The converse is not always true: examples of connected spaces that arenot path-connected include the extended long line L* and the topologists sine curve.However, subsets of the real lineR are connected if and only if they are path-connected; these subsets are the intervalsofR. Also, open subsets ofRn orCn are connected if and only if they are path-connected. Additionally, connectednessand path-connectedness are the same for nite topological spaces.

2.6 Products of spacesMain article: Product topology

Given X such that

X :=Yi2I

Xi;

is the Cartesian product of the topological spaces Xi, indexed by i 2 I , and the canonical projections pi : X Xi,the product topology on X is dened as the coarsest topology (i.e. the topology with the fewest open sets) for whichall the projections pi are continuous. The product topology is sometimes called the Tychono topology.The open sets in the product topology are unions (nite or innite) of sets of the form Qi2I Ui , where each Ui isopen in Xi and Ui Xi only nitely many times. In particular, for a nite product (in particular, for the product oftwo topological spaces), the products of base elements of the Xi gives a basis for the productQi2I Xi .

2.7. SEPARATION AXIOMS 11

The product topology on X is the topology generated by sets of the form pi1(U), where i is in I and U is an opensubset of Xi. In other words, the sets {pi1(U)} form a subbase for the topology on X. A subset of X is open ifand only if it is a (possibly innite) union of intersections of nitely many sets of the form pi1(U). The pi1(U) aresometimes called open cylinders, and their intersections are cylinder sets.In general, the product of the topologies of each Xi forms a basis for what is called the box topology on X. In general,the box topology is ner than the product topology, but for nite products they coincide.Related to compactness is Tychonos theorem: the (arbitrary) product of compact spaces is compact.

2.7 Separation axiomsMain article: Separation axiom

Many of these names have alternative meanings in some of mathematical literature, as explained on History of theseparation axioms; for example, the meanings of normal and T4" are sometimes interchanged, similarly regularand T3", etc. Many of the concepts also have several names; however, the one listed rst is always least likely to beambiguous.Most of these axioms have alternative denitions with the samemeaning; the denitions given here fall into a consistentpattern that relates the various notions of separation dened in the previous section. Other possible denitions can befound in the individual articles.In all of the following denitions, X is again a topological space.

X is T0, or Kolmogorov, if any two distinct points in X are topologically distinguishable. (It is a common themeamong the separation axioms to have one version of an axiom that requires T0 and one version that doesn't.)

X is T1, or accessible or Frchet, if any two distinct points in X are separated. Thus, X is T1 if and onlyif it is both T0 and R0. (Though you may say such things as T1 space, Frchet topology, and Suppose that thetopological spaceX is Frchet, avoid saying Frchet space in this context, since there is another entirely dierentnotion of Frchet space in functional analysis.)

X is Hausdor, or T2 or separated, if any two distinct points in X are separated by neighbourhoods. Thus, Xis Hausdor if and only if it is both T0 and R1. A Hausdor space must also be T1.

X is T2, or Urysohn, if any two distinct points in X are separated by closed neighbourhoods. A T spacemust also be Hausdor.

X is regular, or T3, if it is T0 and if given any point x and closed set F in X such that x does not belong to F,they are separated by neighbourhoods. (In fact, in a regular space, any such x and F is also separated by closedneighbourhoods.)

X is Tychono, or T3, completely T3, or completely regular, if it is T0 and if f, given any point x and closedset F in X such that x does not belong to F, they are separated by a continuous function.

X is normal, orT4, if it is Hausdor and if any two disjoint closed subsets ofX are separated by neighbourhoods.(In fact, a space is normal if and only if any two disjoint closed sets can be separated by a continuous function;this is Urysohns lemma.)

X is completely normal, or T5 or completely T4, if it is T1. and if any two separated sets are separated byneighbourhoods. A completely normal space must also be normal.

X is perfectly normal, or T6 or perfectly T4, if it is T1 and if any two disjoint closed sets are precisely separatedby a continuous function. A perfectly normal Hausdor space must also be completely normal Hausdor.

The Tietze extension theorem: In a normal space, every continuous real-valued function dened on a closed subspacecan be extended to a continuous map dened on the whole space.

12 CHAPTER 2. GENERAL TOPOLOGY

2.8 Countability axiomsMain article: axiom of countability

An axiom of countability is a property of certain mathematical objects (usually in a category) that requires theexistence of a countable set with certain properties, while without it such sets might not exist.Important countability axioms for topological spaces:

sequential space: a set is open if every sequence convergent to a point in the set is eventually in the set rst-countable space: every point has a countable neighbourhood basis (local base) second-countable space: the topology has a countable base separable space: there exists a countable dense subspace Lindelf space: every open cover has a countable subcover -compact space: there exists a countable cover by compact spaces

Relations:

Every rst countable space is sequential. Every second-countable space is rst-countable, separable, and Lindelf. Every -compact space is Lindelf. A metric space is rst-countable. For metric spaces second-countability, separability, and the Lindelf property are all equivalent.

2.9 Metric spacesMain article: Metric space

Ametric space[7] is an ordered pair (M;d) whereM is a set and d is a metric onM , i.e., a function

d : M M ! R

such that for any x; y; z 2M , the following holds:

1. d(x; y) 0 (non-negative),2. d(x; y) = 0 i x = y (identity of indiscernibles),

3. d(x; y) = d(y; x) (symmetry) and

4. d(x; z) d(x; y) + d(y; z) (triangle inequality) .

The function d is also called distance function or simply distance. Often, d is omitted and one just writes M for ametric space if it is clear from the context what metric is used.Every metric space is paracompact and Hausdor, and thus normal.The metrization theorems provide necessary and sucient conditions for a topology to come from a metric.

2.10. BAIRE CATEGORY THEORY 13

2.10 Baire category theoryMain article: Baire category theorem

The Baire category theorem says: If X is a complete metric space or a locally compact Hausdor space, then theinterior of every union of countably many nowhere dense sets is empty.[8]

Any open subspace of a Baire space is itself a Baire space.

2.11 Main areas of research

Three iterations of a Peano curve construction, whose limit is a space-lling curve. The Peano curve is studied in continuum theory,a branch of general topology.

2.11.1 Continuum theoryMain article: Continuum theory

A continuum (pl continua) is a nonempty compact connected metric space, or less frequently, a compact connectedHausdor space. Continuum theory is the branch of topology devoted to the study of continua.

2.11.2 Pointless topologyMain article: Pointless topology

Pointless topology (also called point-free or pointfree topology) is an approach to topology that avoids mentioningpoints. The name 'pointless topology' is due to John von Neumann.[9] The ideas of pointless topology are closelyrelated to mereotopologies, in which regions (sets) are treated as foundational without explicit reference to underlyingpoint sets.

2.11.3 Dimension theoryMain article: Dimension theory

Dimension theory is a branch of general topology dealing with dimensional invariants of topological spaces.

2.11.4 Topological algebrasMain article: Topological algebra

14 CHAPTER 2. GENERAL TOPOLOGY

A topological algebra A over a topological eld K is a topological vector space together with a continuous multipli-cation

: AA ! A(a; b) 7! a bthat makes it an algebra over K. A unital associative topological algebra is a topological ring.The term was coined by David van Dantzig; it appears in the title of his doctoral dissertation (1931).

2.11.5 Metrizability theoryMain article: Metrization theorem

In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to ametric space. That is, a topological space (X; ) is said to be metrizable if there is a metric

d : X X ! [0;1)such that the topology induced by d is . Metrization theorems are theorems that give sucient conditions for atopological space to be metrizable.

2.11.6 Set-theoretic topologyMain article: Set-theoretic topology

Set-theoretic topology is a subject that combines set theory and general topology. It focuses on topological questionsthat are independent of ZermeloFraenkel set theory(ZFC). A famous problem is the normal Moore space question, aquestion in general topology that was the subject of intense research. The answer to the normal Moore space questionwas eventually proved to be independent of ZFC.

2.12 See also List of examples in general topology Glossary of general topology for detailed denitions List of general topology topics for related articles Category of topological spaces

2.13 References[1] Munkres, James R. Topology. Vol. 2. Upper Saddle River: Prentice Hall, 2000.

[2] Adams, Colin Conrad, and Robert David Franzosa. Introduction to topology: pure and applied. Pearson Prentice Hall,2008.

[3] Merrield, Richard E.; Simmons, Howard E. (1989). Topological Methods in Chemistry. New York: John Wiley & Sons.p. 16. ISBN 0-471-83817-9. Retrieved 27 July 2012. Denition. A collection B of subsets of a topological space (X,T)is called a basis for T if every open set can be expressed as a union of members of B.

[4] Armstrong, M. A. (1983). Basic Topology. Springer. p. 30. ISBN 0-387-90839-0. Retrieved 13 June 2013. Suppose wehave a topology on a set X, and a collection of open sets such that every open set is a union of members of . Then is called a base for the topology...

2.14. FURTHER READING 15

[5] Moore, E. H.; Smith, H. L. (1922). A General Theory of Limits. American Journal of Mathematics 44 (2): 102121.doi:10.2307/2370388. JSTOR 2370388

[6] Heine, E.. Die Elemente der Functionenlehre.. Journal fr die reine und angewandte Mathematik 74 (1872): 172-188..

[7] Maurice Frchet introduced metric spaces in his work Sur quelques points du calcul fonctionnel, Rendic. Circ. Mat.Palermo 22 (1906) 174.

[8] R. Baire. Sur les fonctions de variables relles. Ann. di Mat., 3:1123, 1899.

[9] Garrett Birkho, VON NEUMANN AND LATTICE THEORY, John Von Neumann 1903-1957, J. C. Oxtoley, B. J. Pettis,American Mathematical Soc., 1958, page 50-5

2.14 Further readingSome standard books on general topology include:

Bourbaki, Topologie Gnrale (General Topology), ISBN 0-387-19374-X. John L. Kelley (1955) General Topology, link from Internet Archive, originally published by David Van Nos-trand Company.

Stephen Willard, General Topology, ISBN 0-486-43479-6. James Munkres, Topology, ISBN 0-13-181629-2. George F. Simmons, Introduction to Topology and Modern Analysis, ISBN 1-575-24238-9. Paul L. Shick, Topology: Point-Set and Geometric, ISBN 0-470-09605-5. Ryszard Engelking, General Topology, ISBN 3-88538-006-4. Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 507446

O.Ya. Viro, O.A. Ivanov, V.M. Kharlamov and N.Yu. Netsvetaev, Elementary Topology: Textbook in Prob-lems, ISBN 978-0-8218-4506-6.

The arXiv subject code is math.GN.

Chapter 3

Nested intervals

0

0

0

0

In mathematics, a sequence of nested intervals is understood as a collection of sets of real numbers

In

such that each set In is an interval of the real line, for n = 1, 2, 3, ..., and that further

In is a subset of In

for all n. In other words, the intervals diminish, with the left-hand end moving only towards the right, and the right-hand end only to the left.The main question to be posed is the nature of the intersection of all the In. Without any further information, all thatcan be said is that the intersection J of all the In, i.e. the set of all points common to the intervals, is either the emptyset, a point, or some interval.The possibility of an empty intersection can be illustrated by the intersection when In is the open interval

(0, 2n).

Here the intersection is empty, because no number x is both greater than 0 and less than every fraction 2n.The situation is dierent for closed intervals. The nested intervals theorem states that if each In is a closed and boundedinterval, say

In = [an, bn]

with

16

3.1. HIGHER DIMENSIONS 17

an bn

then under the assumption of nesting, the intersection of the In is not empty. It may be a singleton set {c}, or anotherclosed interval [a, b]. More explicitly, the requirement of nesting means that

an an

and

bn bn .

Moreover, if the length of the intervals converges to 0, then the intersection of the In is a singleton.One can consider the complement of each interval, written as (1; an) [ (bn;1) . By De Morgans laws, thecomplement of the intersection is a union of two disjoint open sets. By the connectedness of the real line there mustbe something between them. This shows that the intersection of (even an uncountable number of) nested, closed, andbounded intervals is nonempty.

3.1 Higher dimensionsIn two dimensions there is a similar result: nested closed disks in the plane must have a common intersection. Thisresult was shown by Hermann Weyl to classify the singular behaviour of certain dierential equations.

3.2 See also Bisection Cantors Intersection Theorem

3.3 References Fridy, J. A. (2000), 3.3 The Nested Intervals Theorem, Introductory Analysis: The Theory of Calculus,Academic Press, p. 29, ISBN 9780122676550.

Shilov, Georgi E. (2012), 1.8 The Principle of Nested Intervals, Elementary Real and Complex Analysis,Dover Books on Mathematics, Courier Dover Publications, pp. 2122, ISBN 9780486135007.

Sohrab, Houshang H. (2003), Theorem 2.1.5 (Nested Intervals Theorem)", Basic Real Analysis, Springer, p.45, ISBN 9780817642112.

Chapter 4

Tychonos theorem

For other theorems named after Tychono, see Tychonos theorem (disambiguation).

In mathematics, Tychonos theorem states that the product of any collection of compact topological spaces iscompact with respect to the product topology. The theorem is named after Andrey Nikolayevich Tychono, whoproved it rst in 1930 for powers of the closed unit interval and in 1935 stated the full theorem along with the remarkthat its proof was the same as for the special case. The earliest known published proof is contained in a 1937 paperof Eduard ech.Several texts identify Tychonos theorem as the single most important result in general topology [e.g. Willard, p.120]; others allow it to share this honor with Urysohns lemma.

4.1 Topological denitionsThe theorem depends crucially upon the precise denitions of compactness and of the product topology; in fact,Tychonos 1935 paper denes the product topology for the rst time. Conversely, part of its importance is to givecondence that these particular denitions are the correct (i.e., most useful) ones.Indeed, the HeineBorel denition of compactness that every covering of a space by open sets admits a nitesubcovering is relatively recent. More popular in the 19th and early 20th centuries was the BolzanoWeierstrasscriterion that every sequence admits a convergent subsequence, now called sequential compactness. These conditionsare equivalent for metrizable spaces, but neither one implies the other in the class of all topological spaces.It is almost trivial to prove that the product of two sequentially compact spaces is sequentially compact one passesto a subsequence for the rst component and then a subsubsequence for the second component. An only slightly moreelaborate diagonalization argument establishes the sequential compactness of a countable product of sequentiallycompact spaces. However, the product of continuum many copies of the closed unit interval (with its usual topology)fails to be sequentially compact with respect to the product topology, even though it is compact by Tychonos theorem(e.g., see Wilansky 1970, p. 134).This is a critical failure: if X is a completely regular Hausdor space, there is a natural embedding from X into[0,1]C(X,[0,1]), where C(X,[0,1]) is the set of continuous maps from X to [0,1]. The compactness of [0,1]C(X,[0,1]) thusshows that every completely regular Hausdor space embeds in a compact Hausdor space (or, can be compacti-ed.) This construction is the Stoneech compactication. Conversely, all subspaces of compact Hausdor spacesare completely regular Hausdor, so this characterizes the completely regular Hausdor spaces as those that can becompactied. Such spaces are now called Tychono spaces.

4.2 ApplicationsTychonos theorem has been used to prove many other mathematical theorems. These include theorems aboutcompactness of certain spaces such as the BanachAlaoglu theorem on the weak-* compactness of the unit ball ofthe dual space of a normed vector space, and the ArzelAscoli theorem characterizing the sequences of functions

18

4.3. PROOFS OF TYCHONOFFS THEOREM 19

in which every subsequence has a uniformly convergent subsequence. They also include statements less obviouslyrelated to compactness, such as the De BruijnErds theorem stating that every minimal k-chromatic graph is nite,and the CurtisHedlundLyndon theorem providing a topological characterization of cellular automata.As a rule of thumb, any sort of construction that takes as input a fairly general object (often of an algebraic, ortopological-algebraic nature) and outputs a compact space is likely to use Tychono: e.g., the Gelfand space ofmaximal ideals of a commutative C* algebra, the Stone space of maximal ideals of a Boolean algebra, and theBerkovich spectrum of a commutative Banach ring.

4.3 Proofs of Tychonos theorem1) Tychonos 1930 proof used the concept of a complete accumulation point.2) The theorem is a quick corollary of the Alexander subbase theorem.More modern proofs have been motivated by the following considerations: the approach to compactness via con-vergence of subsequences leads to a simple and transparent proof in the case of countable index sets. However, theapproach to convergence in a topological space using sequences is sucient when the space satises the rst axiomof countability (as metrizable spaces do), but generally not otherwise. However, the product of uncountably manymetrizable spaces, each with at least two points, fails to be rst countable. So it is natural to hope that a suitablenotion of convergence in arbitrary spaces will lead to a compactness criterion generalizing sequential compactness inmetrizable spaces that will be as easily applied to deduce the compactness of products. This has turned out to be thecase.3) The theory of convergence via lters, due to Henri Cartan and developed by Bourbaki in 1937, leads to thefollowing criterion: assuming the ultralter lemma, a space is compact if and only if each ultralter on the spaceconverges. With this in hand, the proof becomes easy: the (lter generated by the) image of an ultralter on theproduct space under any projection map is an ultralter on the factor space, which therefore converges, to at least onexi. One then shows that the original ultralter converges to x = (xi). In his textbook, Munkres gives a reworking ofthe CartanBourbaki proof that does not explicitly use any lter-theoretic language or preliminaries.4) Similarly, the MooreSmith theory of convergence via nets, as supplemented by Kelleys notion of a universal net,leads to the criterion that a space is compact if and only if each universal net on the space converges. This criterionleads to a proof (Kelley, 1950) of Tychonos theorem, which is, word for word, identical to the Cartan/Bourbakiproof using lters, save for the repeated substitution of universal net for ultralter base.5) A proof using nets but not universal nets was given in 1992 by Paul Cherno.

4.4 Tychonos theorem and the axiom of choiceAll of the above proofs use the axiom of choice (AC) in some way. For instance, the third proof uses that every lteris contained in an ultralter (i.e., a maximal lter), and this is seen by invoking Zorns lemma. Zorns lemma is alsoused to prove Kelleys theorem, that every net has a universal subnet. In fact these uses of AC are essential: in 1950Kelley proved that Tychonos theorem implies the axiom of choice. Note that one formulation of AC is that theCartesian product of a family of nonempty sets is nonempty; but since the empty set is most certainly compact, theproof cannot proceed along such straightforward lines. Thus Tychonos theorem joins several other basic theorems(e.g. that every nonzero vector space has a basis) in being equivalent to AC.On the other hand, the statement that every lter is contained in an ultralter does not imply AC. Indeed, it is nothard to see that it is equivalent to the Boolean prime ideal theorem (BPI), a well-known intermediate point betweenthe axioms of Zermelo-Fraenkel set theory (ZF) and the ZF theory augmented by the axiom of choice (ZFC). A rstglance at the second proof of Tychno may suggest that the proof uses no more than (BPI), in contradiction to theabove. However, the spaces in which every convergent lter has a unique limit are precisely the Hausdor spaces. Ingeneral we must select, for each element of the index set, an element of the nonempty set of limits of the projectedultralter base, and of course this uses AC. However, it also shows that the compactness of the product of compactHausdor spaces can be proved using (BPI), and in fact the converse also holds. Studying the strength of Tychonostheorem for various restricted classes of spaces is an active area in set-theoretic topology.The analogue of Tychonos theorem in pointless topology does not require any form of the axiom of choice.

20 CHAPTER 4. TYCHONOFFS THEOREM

4.5 Proof of the axiom of choice from Tychonos theoremTo prove that Tychonos theorem in its general version implies the axiom of choice, we establish that every innitecartesian product of non-empty sets is nonempty. The trickiest part of the proof is introducing the right topology.The right topology, as it turns out, is the conite topology with a small twist. It turns out that every set given thistopology automatically becomes a compact space. Once we have this fact, Tychonos theorem can be applied; wethen use the nite intersection property (FIP) denition of compactness. The proof itself (due to J. L. Kelley) follows:Let {Ai} be an indexed family of nonempty sets, for i ranging in I (where I is an arbitrary indexing set). We wishto show that the cartesian product of these sets is nonempty. Now, for each i, take Xi to be Ai with the index i itselftacked on (renaming the indices using the disjoint union if necessary, we may assume that i is not a member of Ai,so simply take Xi = Ai {i}).Now dene the cartesian product

X =Yi2I

Xi

along with the natural projection maps i which take a member of X to its ith term.We give each Xi the topology whose open sets are the conite subsets of Xi, plus the empty set (the conite topology)and the singleton {i}. This makes Xi compact, and by Tychonos theorem, X is also compact (in the producttopology). The projection maps are continuous; all the Ai's are closed, being complements of the singleton open set{i} in Xi. So the inverse images i1(Ai) are closed subsets of X. We note that

Yi2I

Ai =\i2I

1i (Ai)

and prove that these inverse images are nonempty and have the FIP. Let i1, ..., iN be a nite collection of indicesin I. Then the nite product Ai1 ... AiN is non-empty (only nitely many choices here, so AC is not needed); itmerely consists of N-tuples. Let a = (a1, ..., aN) be such an N-tuple. We extend a to the whole index set: take a tothe function f dened by f(j) = ak if j = ik, and f(j) = j otherwise. This step is where the addition of the extra point toeach space is crucial, for it allows us to dene f for everything outside of the N-tuple in a precise way without choices(we can already choose, by construction, j from Xj ). ik(f) = ak is obviously an element of each Aik so that f is ineach inverse image; thus we have

N\k=1

1ik (Aik) 6= ?:

By the FIP denition of compactness, the entire intersection over I must be nonempty, and the proof is complete.

4.6 References Cherno, Paul N. (1992), A simple proof of Tychonos theorem via nets, American Mathematical Monthly99 (10): 932934, doi:10.2307/2324485, JSTOR 2324485.

Johnstone, Peter T. (1982), Stone spaces, Cambridge Studies in Advanced Mathematics 3, New York: Cam-bridge University Press, ISBN 0-521-23893-5.

Johnstone, Peter T. (1981), Tychonos theorem without the axiom of choice, Fundamenta Mathematica113: 2135.

Kelley, John L. (1950), Convergence in topology,DukeMathematical Journal 17 (3): 277283, doi:10.1215/S0012-7094-50-01726-1.

Kelley, John L. (1950), The Tychono product theorem implies the axiom of choice, Fundamenta Mathe-matica 37: 7576.

4.7. EXTERNAL LINKS 21

Munkres, James (2000), Topology (2nd ed.), Upper Saddle River, NJ: Prentice Hall, ISBN 0-13-181629-2. Tychono, Andrey N. (1930), "ber die topologische Erweiterung von Rumen,Mathematische Annalen (inGerman) 102 (1): 544561, doi:10.1007/BF01782364.

Wilansky, A. (1970), Topology for Analysis, Ginn and Company Willard, Stephen (2004), General Topology, Mineola, NY: Dover Publications, ISBN 0-486-43479-6.

4.7 External links Tychonos Theorem at ProofWiki

Chapter 5

Uncountable set

Uncountable redirects here. For the linguistic concept, see Uncountable noun.

In mathematics, an uncountable set (or uncountably innite set)[1] is an innite set that contains too many elementsto be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinalnumber is larger than that of the set of all natural numbers.

5.1 CharacterizationsThere are many equivalent characterizations of uncountability. A set X is uncountable if and only if any of thefollowing conditions holds:

There is no injective function from X to the set of natural numbers. X is nonempty and every -sequence of elements of X fails to include at least one element of X. That is, X isnonempty and there is no surjective function from the natural numbers to X.

The cardinality of X is neither nite nor equal to @0 (aleph-null, the cardinality of the natural numbers). The set X has cardinality strictly greater than @0 .

The rst three of these characterizations can be proven equivalent in ZermeloFraenkel set theory without the axiomof choice, but the equivalence of the third and fourth cannot be proved without additional choice principles.

5.2 Properties If an uncountable set X is a subset of set Y, then Y is uncountable.

5.3 ExamplesThe best known example of an uncountable set is the set R of all real numbers; Cantors diagonal argument showsthat this set is uncountable. The diagonalization proof technique can also be used to show that several other sets areuncountable, such as the set of all innite sequences of natural numbers and the set of all subsets of the set of naturalnumbers. The cardinality of R is often called the cardinality of the continuum and denoted by c, or 2@0 , or i1(beth-one).The Cantor set is an uncountable subset of R. The Cantor set is a fractal and has Hausdor dimension greater thanzero but less than one (R has dimension one). This is an example of the following fact: any subset of R of Hausdordimension strictly greater than zero must be uncountable.

22

5.4. WITHOUT THE AXIOM OF CHOICE 23

Another example of an uncountable set is the set of all functions from R to R. This set is even more uncountablethan R in the sense that the cardinality of this set is i2 (beth-two), which is larger than i1 .A more abstract example of an uncountable set is the set of all countable ordinal numbers, denoted by or 1.The cardinality of is denoted @1 (aleph-one). It can be shown, using the axiom of choice, that @1 is the smallestuncountable cardinal number. Thus either i1 , the cardinality of the reals, is equal to @1 or it is strictly larger. GeorgCantor was the rst to propose the question of whether i1 is equal to @1 . In 1900, David Hilbert posed this questionas the rst of his 23 problems. The statement that @1 = i1 is now called the continuum hypothesis and is known tobe independent of the ZermeloFraenkel axioms for set theory (including the axiom of choice).

5.4 Without the axiom of choiceWithout the axiom of choice, theremight exist cardinalities incomparable to@0 (namely, the cardinalities ofDedekind-nite innite sets). Sets of these cardinalities satisfy the rst three characterizations above but not the fourth charac-terization. Because these sets are not larger than the natural numbers in the sense of cardinality, some may not wantto call them uncountable.If the axiom of choice holds, the following conditions on a cardinal are equivalent:

@0; > @0; and @1 , where @1 = j!1j and !1 is least initial ordinal greater than !:

However, these may all be dierent if the axiom of choice fails. So it is not obvious which one is the appropriategeneralization of uncountability when the axiom fails. It may be best to avoid using the word in this case and specifywhich of these one means.

5.5 See also Aleph number Beth number Injective function Natural number

5.6 References[1] Uncountably Innite from Wolfram MathWorld

Halmos, Paul, Naive Set Theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition). Reprinted by Martino Fine Books,2011. ISBN 978-1-61427-131-4 (Paperback edition).

Jech, Thomas (2002), Set Theory, Springer Monographs in Mathematics (3rd millennium ed.), Springer, ISBN3-540-44085-2

5.7 External links Proof that R is uncountable

24 CHAPTER 5. UNCOUNTABLE SET

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Finite intersection propertyDefinitionDiscussionApplicationsExamplesTheorems VariantsReferences

General topologyHistory A topology on a setBasis for a topologySubspace and quotientExamples of topological spaces

Continuous functionsAlternative definitionsPropertiesHomeomorphismsDefining topologies via continuous functions

Compact setsConnected setsConnected componentsDisconnected spacesPath-connected sets

Products of spacesSeparation axiomsCountability axiomsMetric spacesBaire category theoryMain areas of researchContinuum theoryPointless topologyDimension theoryTopological algebrasMetrizability theorySet-theoretic topology

See also References Further reading

Nested intervalsHigher dimensionsSee alsoReferences

Tychonoffs theoremTopological definitions Applications Proofs of Tychonoffs theorem Tychonoffs theorem and the axiom of choice Proof of the axiom of choice from Tychonoffs theoremReferences External links

Uncountable setCharacterizationsPropertiesExamples Without the axiom of choiceSee alsoReferences External linksText and image sources, contributors, and licensesTextImagesContent license