GENEL TOPOLOJİ - Başkent...

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GENEL TOPOLOJİ Timur Karaçay Dizgi Düzeltmeleri (15 Eylül 2010)

Transcript of GENEL TOPOLOJİ - Başkent...

Page 1: GENEL TOPOLOJİ - Başkent Üniversitesimail.baskent.edu.tr/~tkaracay/etudio/ders/math/topology/... · 2017. 3. 26. · X !"# Y $#%$&'(! !)! )*+# ,"-.'/ X )*+#-!'0#' Y )*+#-!'# 1&'2+"2

GENEL

TOPOLOJİ Timur Karaçay

Dizgi Düzeltmeleri

(15 Eylül 2010)

Page 2: GENEL TOPOLOJİ - Başkent Üniversitesimail.baskent.edu.tr/~tkaracay/etudio/ders/math/topology/... · 2017. 3. 26. · X !"# Y $#%$&'(! !)! )*+# ,"-.'/ X )*+#-!'0#' Y )*+#-!'# 1&'2+"2

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T =⋃{Vx : x ∈ T}

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Page 15: GENEL TOPOLOJİ - Başkent Üniversitesimail.baskent.edu.tr/~tkaracay/etudio/ders/math/topology/... · 2017. 3. 26. · X !"# Y $#%$&'(! !)! )*+# ,"-.'/ X )*+#-!'0#' Y )*+#-!'# 1&'2+"2

! !"#$ %& '#()*"+ ,-.*'+/-."0(

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f−1(W ) ∈ B(x0) 342E2'(1&, U = f−1(W ) 241&G2' B2D /1/ F2&4151? =%*%/

f(U) = ff−1(W ) = W ⊂ V A251/(1G1/.2/ 01'2&,

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Page 16: GENEL TOPOLOJİ - Başkent Üniversitesimail.baskent.edu.tr/~tkaracay/etudio/ders/math/topology/... · 2017. 3. 26. · X !"# Y $#%$&'(! !)! )*+# ,"-.'/ X )*+#-!'0#' Y )*+#-!'# 1&'2+"2

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Page 17: GENEL TOPOLOJİ - Başkent Üniversitesimail.baskent.edu.tr/~tkaracay/etudio/ders/math/topology/... · 2017. 3. 26. · X !"# Y $#%$&'(! !)! )*+# ,"-.'/ X )*+#-!'0#' Y )*+#-!'# 1&'2+"2

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Page 18: GENEL TOPOLOJİ - Başkent Üniversitesimail.baskent.edu.tr/~tkaracay/etudio/ders/math/topology/... · 2017. 3. 26. · X !"# Y $#%$&'(! !)! )*+# ,"-.'/ X )*+#-!'0#' Y )*+#-!'# 1&'2+"2

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Page 19: GENEL TOPOLOJİ - Başkent Üniversitesimail.baskent.edu.tr/~tkaracay/etudio/ders/math/topology/... · 2017. 3. 26. · X !"# Y $#%$&'(! !)! )*+# ,"-.'/ X )*+#-!'0#' Y )*+#-!'# 1&'2+"2

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Page 20: GENEL TOPOLOJİ - Başkent Üniversitesimail.baskent.edu.tr/~tkaracay/etudio/ders/math/topology/... · 2017. 3. 26. · X !"# Y $#%$&'(! !)! )*+# ,"-.'/ X )*+#-!'0#' Y )*+#-!'# 1&'2+"2

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92,B-%-<- :#)1+)7

Page 29: GENEL TOPOLOJİ - Başkent Üniversitesimail.baskent.edu.tr/~tkaracay/etudio/ders/math/topology/... · 2017. 3. 26. · X !"# Y $#%$&'(! !)! )*+# ,"-.'/ X )*+#-!'0#' Y )*+#-!'# 1&'2+"2

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Page 30: GENEL TOPOLOJİ - Başkent Üniversitesimail.baskent.edu.tr/~tkaracay/etudio/ders/math/topology/... · 2017. 3. 26. · X !"# Y $#%$&'(! !)! )*+# ,"-.'/ X )*+#-!'0#' Y )*+#-!'# 1&'2+"2

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!$ n ∈ Z 7.*,+ (9323 (n, n+ 1) ,2,./+.,2/'/' C4.3)4*4 ,</+6/2$ B&16324'49$

D$

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Page 31: GENEL TOPOLOJİ - Başkent Üniversitesimail.baskent.edu.tr/~tkaracay/etudio/ders/math/topology/... · 2017. 3. 26. · X !"# Y $#%$&'(! !)! )*+# ,"-.'/ X )*+#-!'0#' Y )*+#-!'# 1&'2+"2

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B$ Rn23045,5, R 23045,0 &74085)5 9:;<&;<;'8:-=> ;)<0C5?5,5 @A+1&'-,-3$

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40,- f(x) = x ;)0, 6-' x ∈ [0, 1] ;)C2?2,2 @A+1&'-,-3$ [0, 1] *080)5 0'0)5?54&'-,& [0, 1) 40 C0 (0, 1] 40'5G0(5* 0'0)5*)0'5 *;,2)2'+0 ,& ;)2'H

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A = {∪Aλ, λ ∈ Λ}

Page 32: GENEL TOPOLOJİ - Başkent Üniversitesimail.baskent.edu.tr/~tkaracay/etudio/ders/math/topology/... · 2017. 3. 26. · X !"# Y $#%$&'(! !)! )*+# ,"-.'/ X )*+#-!'0#' Y )*+#-!'# 1&'2+"2

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Page 33: GENEL TOPOLOJİ - Başkent Üniversitesimail.baskent.edu.tr/~tkaracay/etudio/ders/math/topology/... · 2017. 3. 26. · X !"# Y $#%$&'(! !)! )*+# ,"-.'/ X )*+#-!'0#' Y )*+#-!'# 1&'2+"2

!" !"#$ %&' $()*+, -./0"/*

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x → ‖x‖2 = >: ?@

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3$ 4!567286! $9!&8!("!6"$%!5!5 (ℓ2, ‖.‖) 2+345)-3 -3 9&B&,7( 67-282)2 91'0&,C

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2

+

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2

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43M573.(7(,:

Page 34: GENEL TOPOLOJİ - Başkent Üniversitesimail.baskent.edu.tr/~tkaracay/etudio/ders/math/topology/... · 2017. 3. 26. · X !"# Y $#%$&'(! !)! )*+# ,"-.'/ X )*+#-!'0#' Y )*+#-!'# 1&'2+"2

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<1*1 L2[0, 1] ).$ '/&+$,$.)8= >( (4-< 34$,)*5$

ϕ → ‖ϕ‖2 =

(∫

1

0

‖ϕ(x)‖2dx

)

1

2

? @=ABC

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'/+3,$* 5/*39383* %), *;,8 ;.5(2(*( '/&+$,)*)4=

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