Normal space goes to Normal space.

by MathematicalPhysicist
Tags: normal, space
MathematicalPhysicist is offline
Feb1-08, 02:10 PM
P: 3,175
here's the question:
prove that if: f:X->Y is onto Y, closed (i.e return closed sets from given closed sets in X) and continuous then if X is Normal (satisfy axioms: T1 and T4) then also Y is Normal.
Now I've showed that if X is T1 then Y is T1, but I'm having difficulty with T4.
here's what I did:
let F,G be disjoint closed sets of Y, then by continuity f^-1(F) and f^-1(G) are closed in X, and they are disjoint because: f^-1(FחG)=f^-1(G)חf^-1(F), now because X is T4 we have neighbourhoods of f^-1(F) and f^-1(G) which are disjoint, now I need to show that also F a G have this property, I guess I need to use the onto feature, but how?

any hints?
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gel is offline
Feb1-08, 04:44 PM
gel's Avatar
P: 532
Your neighbourhoods of f^-1(F) and f^-1(G) can be chosen to be open. Take their complements, apply f to get two closed sets in Y, then take their complements and show that these open sets separate F and G.
MathematicalPhysicist is offline
Feb2-08, 02:23 AM
P: 3,175
you mean, f^-1(F), and f^-1(G) are contained in U_G and U_F which are open and disjoint in X, and then apply the complement, and then apply f, ok that's what i did before but it got me to nowhere, i.e
f(X-U_G) closed and contained in f(X-f^-1(G), and the same with F just change the G with F, then I take the complement wrt Y, but now I need to show that:
G is contained in Y-f(X-f^-1(G)) (the same for F), but not sure how to do it i mean, if
y is in G and not in Y-f(X-f^-1(G)) then y is in f(X-f^-1(G)) so there's x in X-f^-1(G) s.t y=f(x), but then x isnt in f^-1(G) thus f(x) isnt in G, a contradiction.

ok, i see now, don't know how i got it wrong before... (-:

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