1. ## topology, continuity

Prove the following are equivalent:
(a) $\displaystyle \forall V \subseteq Y$, $\displaystyle V$ is an open subset of $\displaystyle Y$ iff $\displaystyle p^{-1}(V)$ is an open subset of $\displaystyle X$
(b) $\displaystyle F$ a closed subset of $\displaystyle Y$ iff $\displaystyle p^{-1}(F)$ is a closed subset of $\displaystyle X$.

2. Hello,

is stronger than continuity.
what do you mean by this ?

3. I am not sure, we care covering strong continuity, but we didn't go over it a lot, that's why I'm stuck on this one.

4. I am not 100% sure but I think the question (a) asks to show an "quotient map" is stronger than "continuous map".

In (a) "iff" corresponds an "quotient map" and "if" corresponds a "continuous map".

5. I figured it out.

6. I will do one way for you if you promise to show us all the other way.
$\displaystyle a \Rightarrow b$
If $\displaystyle F \subseteq Y\,\& \,F$ is closed then $\displaystyle Y\backslash F$ is open in $\displaystyle Y$.
Therefore, $\displaystyle p^{ - 1} \left( {Y\backslash F} \right)$ is open in $\displaystyle X$.
But $\displaystyle p^{ - 1} \left( {Y\backslash F} \right) = p^{ - 1} \left( Y \right)\backslash p^{ - 1} \left( F \right) = X\backslash p^{ - 1} \left( F \right)$ is open.
Therefore, $\displaystyle p^{ - 1} \left( F \right)$ is closed in $\displaystyle X$.

7. $\displaystyle b \Rightarrow a$, let $\displaystyle V$ be closed in $\displaystyle Y$. $\displaystyle Y - V$ is closed in $\displaystyle Y$, so that $\displaystyle p^{ - 1}(Y - V)$ is closed in $\displaystyle X$. Thus $\displaystyle X - p^{ - 1}(Y - V) = X - (p^{ - 1}(Y) - p^{ - 1}(V)) = X - (X - p^{ - 1}(V)) = p^{ - 1}(V)$ is open in $\displaystyle X$. Every statement made works when starting with $\displaystyle p^{ - 1}(V)$ open in $\displaystyle X$. Q.E.D.