 # 1. Let N be a collection of subsets of X, and let O be the smallest...

## Question

Show transcribed text

## Transcribed Text

1. Let N be a collection of subsets of X, and let O be the smallest topology on X that contains N. a) Show that f: W X is continuous is open in W for every NEN. Hint: Think about the topology P = {U If (U) is open in W}. b) Now let A C X be a subset, and write OA for the subspace topology. Write: N = {ANN M N E N}. Show that OA is the smallest topology on A containing NA. Hint: Use part a) to show that the identity function (of sets) gives continuous maps (A, O) (A, NA) 2. Let X and Y be topological spaces, and suppose UCn, where each Ci is a closed subset of X. Show that a function f: X Y is continuous iff each restriction flc, is continuous. Is it true if the subsets are not closed?

## Solution Preview

These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice. Unethical use is strictly forbidden.

By purchasing this solution you'll be able to access the following files:
Solution.pdf.

\$20.00
for this solution

or FREE if you
register a new account!

PayPal, G Pay, ApplePay, Amazon Pay, and all major credit cards accepted.

### Find A Tutor

View available Topology Tutors

Get College Homework Help.

Are you sure you don't want to upload any files?

Fast tutor response requires as much info as possible.