313-16-05 - Victoria University of Wellington School of Mathematics and Statistics MATH 313 Topology 2016 Assignment 5 Due end of Thursday 14 April 1

# 313-16-05 - Victoria University of Wellington School of...

• Test Prep
• Julia.S
• 1

This preview shows page 1 out of 1 page.

Victoria University of Wellington School of Mathematics and Statistics MATH 313 Topology 2016 Assignment 5 Due: end of Thursday 14 April. 1. Let X and Y be topological spaces. Prove that a function f : X Y is continuous if, and only if, f - 1 ( C ) is a closed subset of X whenever C is a closed subset of Y . [Hint: use the fact that f - 1 preserves set complements. See Assignment 4, Question 3.] [4] 2. Let X and Y be topological spaces, and let S be a subbasis for the topology on Y . Prove that a function f : X Y is continuous if, and only if, f - 1 ( S ) is open in X for every S ∈ S . [4] 3. Let f : X Y be continuous, and A a subspace of X . Let f A be the restriction of f to A . Prove that f A : A Y is continuous when A has the subspace topology inherited from X . [Hint: if V is an open subset of Y , consider the relationship between f - 1 A ( V ) and f - 1 ( V ).] [4] 4. Let X be a topological space, A a subset of X , and i : A X the inclusion function having i ( x ) = x for all x A . Show that the subspace topology on A is the coarsest topology on A

#### You've reached the end of your free preview.

Want to read the whole page?

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern