(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let p : X --> Y be a closed, continuous and surjective map. Show that if X is normal, so is Y.

3. The attempt at a solution

I used the following lemma:

X is normal iff given a closed set A and open set U containing A, there is an open set V containing A and whose closure is contained in U.

So, let A be a closed set in Y, and U some neighborhood of A. By continuity, f^-1(A) is closed in X, and f^-1(U) is open in X. Further on, f^-1(U) is an open neighborhood of f^-1(A), since it contains f^-1(A). If we apply the lemma above to these sets, we can find an open set V which satisfies the criterion, since X is regular. Since p is a closed map, the image of Cl(V) is closed, and since all the inclusions remain preserved, Y is normal.

By the way, just to check, the requirement for p to be surjective was because we have chosen a set in Y, and surjectivity guarantees that this set has a well-defined preimage, right?

Thanks in advance.

**Physics Forums - The Fusion of Science and Community**

# Closed continuous surjective map and normal spaces

Know someone interested in this topic? Share a link to this question via email,
Google+,
Twitter, or
Facebook

Have something to add?

- Similar discussions for: Closed continuous surjective map and normal spaces

Loading...

**Physics Forums - The Fusion of Science and Community**