an exercise about normal spaces

Question

this is an exercise from munkres topology book:

Let $p : X \rightarrow Y$ be a closed continuous surjective map. Then if $X$ is normal, prove that $Y$ is normal.

Munkres has left a hint:

first prove that if $p ^{−1} (y) \subset U$ where $U$ is an open subspace of $X$, then $p ^{−1} (W ) \subset U$ for some neighborhood $W \subset Y$ of $y$.

I've proved the hint, but I dont know how to continue.

help pls!

thank u!

http://math.stackexchange.com/questions/276340/image-of-a-normal-space-under-a-closed-and-continuous-map-is-normal — The very fluffy Panda, May 08 '14 at 18:43
@ Panda Bear no! ive seen that link.but there is no answer for my question there. — user115608, May 08 '14 at 18:56
no!it is just about the hint that ive proved it not about my question — user115608, May 08 '14 at 19:05
I posted that because the question is the same and it provides you some more hints. It wasn't meant to be a direct answer. — The very fluffy Panda, May 08 '14 at 19:11

score 2 · Answer 1 · answered May 08 '14 at 19:05

If $A$ and $B$ are closed disjoint subsets of $Y$, then $A'=p^{-1}(A)$ and $B'=p^{-1}(B)$ are closed disjoint subsets of $X$, so they have disjoint open neighborhoods $U\supseteq A'$ and $V⊇B'$. Applying the hint, for each point $a\in A$, there's an open set $U_a$ containing $a$ so that $p^{-1}(U_a)\subseteq U$. Their union $\bigcup_A U_a$ is an open set around $A$. Can you finish it from here?

an exercise about normal spaces

1 Answers1