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by: Rae Kutch

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# Intro Anlysis & Topology MATH 381

Rae Kutch
WVU
GPA 3.8

Staff

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## Popular in Mathematics (M)

This 2 page Class Notes was uploaded by Rae Kutch on Saturday September 12, 2015. The Class Notes belongs to MATH 381 at West Virginia University taught by Staff in Fall. Since its upload, it has received 9 views. For similar materials see /class/202658/math-381-west-virginia-university in Mathematics (M) at West Virginia University.

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Date Created: 09/12/15
Exercises on Open Sets 31 Let X be a topological space with basis 3 and topology T Prove a set 0 Q X is open if and only if 0 can be written as a union of elements in 3 Solution Suppose that O is open By de nition of open sets Vz E O 3Nm E B such that z E Nm and Nm Q 0 Let 0 U Nm 160 Then since each Nm Q 0 we have 0 Q 0 On the other hand Vz E O z E Nm Q 0 Thus 0 O and so 0 can be written as a union of elements in 3 Conversely suppose that O can be written as a union of elements in B that is O U Bi where each Bi 6 B we want to apply the de nition to show that O is open Vz E 0 since 0 U Bi there must be some B E B such that z 6 Bi Therefore by the de nition of open sets 0 is open 33 Let X zyz the set with three elements Find out as many topology as you can for X by listing their open sets Solution We can list all the open sets of a topology on X using the fact that T is a collection of open sets of a topology on X if sets in T satisfy the following Cl 0 e T C2 A nite intersection of members in T is also in T CB Any union of of members in T is also in T 171 07X 1072 07 7997 Ts 07 bleL T4 07 Cle in T5 07 avb7 X7 T6 07 avc7X7 T7 07 bvc7X39 Vi T8 07 a7 a7 b7 X7 T9 07 b7 a7 b7 X7 T10 07 C7 a7 b7 X7 T11 07 a7 a7 C7 X7 T12 07 b7 a7 C7 X7 T13 07 C7 a7 C7 X7 T14 07 a7 b7 C7 X7 T15 07 b7 b7 CL X7 T16 0 c b c X V T17 07 a7 b7 a7 b7 X7 T18 07 a7 C7 a7 C7 X7 T19 07 b7 C7 b7 C7 Vi T20 07 a7 a7 b7 a7 C7 X7 T21 07 b7 a7 b7 b7 C7 X7 T22 07 C7 a7 C7 b7 C7 Vii T23 07 a7 b7 a7 b7 a7 C7 X7 T24 07 a7 b7 a7 b7 b7 CL X7 T25 07 a7 b7 a7 b7 a7 C7 X7 T26 07 a7 C7 a7 b7 a7 CL X7 T27 07 a7 C7 a7 b7 b7 C7 X7 T28 07 a7 C7 a7 b7 a7 C7 X7 T29 07 b7 C7 a7 b7 a7 C7 X7 T30 07 b7 C7 a7 b7 b7 CL X7 T31 07 b7 C7 a7 b7 a7 C7 Viii T32 0 a b 0 11 1 0 120 X 35 Let X be a topological space with two bases 3 and B Show that B is equivalent to 3 if and only if both of the following hold i VB 6 B and Vz E B 3B 6 3 such that z E B Q B and ii VB 6 B and Vz E B 3B 6 B such that z E B Q B Proof Let T denote the open sets with bases 3 and T denote the open sets with bases 8 Suppose that B is equivalent to 8 Then T T VB 6 B B is an open set in T T and so by the de nition of open sets in T Vz E B 3B 6 3 such that z E B Q B This proves i Similarly VB 6 B B is an open set in T T and so by the de nition of open sets in T Vz E B 3B 6 B such that z E B Q B This proves ii Conversely we assume and ii to show the two topologies are equivalent V0 6 T by the de nition of open sets in T Vz E 0 3B 6 B x E B Q 0 By i Vz E B 3B 6 3 such that z E B Q B Q 0 Thus Vz E 0 3B 6 8 z E B Q 0 By the de nition of open sets in T O E T This proves that T Q T lnterchanging T and T in the proof above we also have T Q T and so T T Thus the two topologies are equivalent 37 Let A be subset of a topological space X with open sets T Show that each of the following holds i A subset O Q A is open in A iff HO 6 T such that O 0 O A ii A subset Cquot Q A is closed in A iff EC Q X with E 7 C E T such that Cquot C O A Proof ii Suppose that Cquot Q A is closed in A By the de nition of closed sets 0 A 7 Cquot is open in A By i 30 E T such that O 0 O A Let C X 7 0 Then by the de nition ofclosed sets C is closed in X Thus Cquot A7O A7 O A X70 A C A Now suppose that EC Q X with E 7 C E T such that Cquot C O A To show that Cquot Q A is closed in A by de nition of subspace topology we need to show that A 7 Cquot is open in A Let 0 X 7 C Then by the de nition of closed sets 0 is open in X By i O 0 A is open in A NowA7C A7C A A X7C A O O is open in A By the de nition of subspace topology Cquot is closed in A

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