Is Quotient Map Closed? Proof and Explanation | Math Homework

[pivoxa15]

Homework Statement

THe quotient map f is open but is it also closed?

The Attempt at a Solution

I think it is. Consider f: X->Y

FOr every open set V in Y there exists by definition an open set f^-1(V) in X. There is a one to one correspondence between open sets in X and open sets in Y by definition.

So for every closed set V complement in Y there exists a closed set f^-1(V) complement in X. So f is both closed and open.

 

[morphism]

Are you asking whether or not an open quotient map is closed? If so, then what you did does not prove that. There isn't a 1-1 correspondence between open sets in X and open sets in Y. Look carefully at the definition. If f:X->Y is a quotient map, then U is open in Y iff f^-1(U) is open in X. This does not exhaust all the open sets in X. So you haven't proved that f takes closed sets to closed sets.

Anyway, this is false. For a counterexample, let $\pi_1 : \mathbb{R}^2 \to \mathbb{R}$ be the projection map onto the first coordinate. Then $\pi_1$ is an open surjection, so it's a quotient map. However it's not closed. (I'll let you find a closed set that doesn't get mapped to a closed set.)