### Create a StudySoup account

#### Be part of our community, it's free to join!

Already have a StudySoup account? Login here

# Intro Anlysis & Topology MATH 381

WVU

GPA 3.8

### View Full Document

## 9

## 0

## Popular in Course

## 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.

## Reviews for Intro Anlysis & Topology

### What is Karma?

#### Karma is the currency of StudySoup.

#### You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

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

### BOOM! Enjoy Your Free Notes!

We've added these Notes to your profile, click here to view them now.

### You're already Subscribed!

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

## Why people love StudySoup

#### "Knowing I can count on the Elite Notetaker in my class allows me to focus on what the professor is saying instead of just scribbling notes the whole time and falling behind."

#### "I bought an awesome study guide, which helped me get an A in my Math 34B class this quarter!"

#### "There's no way I would have passed my Organic Chemistry class this semester without the notes and study guides I got from StudySoup."

#### "Their 'Elite Notetakers' are making over $1,200/month in sales by creating high quality content that helps their classmates in a time of need."

### Refund Policy

#### STUDYSOUP CANCELLATION POLICY

All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

#### STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com

Satisfaction Guarantee: If you’re not satisfied with your subscription, you can contact us for further help. Contact must be made within 3 business days of your subscription purchase and your refund request will be subject for review.

Please Note: Refunds can never be provided more than 30 days after the initial purchase date regardless of your activity on the site.