Register to reply

Normal space goes to Normal space.

by MathematicalPhysicist
Tags: normal, space
Share this thread:
MathematicalPhysicist
#1
Feb1-08, 02:10 PM
P: 3,243
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?
Phys.Org News Partner Science news on Phys.org
FIXD tells car drivers via smartphone what is wrong
Team pioneers strategy for creating new materials
Team defines new biodiversity metric
gel
#2
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
#3
Feb2-08, 02:23 AM
P: 3,243
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... (-:


Register to reply

Related Discussions
Converting a vector from world space to local space Classical Physics 0
Band diagram in real space vs reciprocal space Atomic, Solid State, Comp. Physics 3
Could any curved space be a cut in a higher-dimensional flat space ? Special & General Relativity 11
Phi- normal distribution (how to look normal tables ) Set Theory, Logic, Probability, Statistics 3