metric space, continuity, open and close

Question

enter image description here Prove that if F,G are closed in X and f, g are continuous, then f ∧ g is continuous.

I know that if I can prove (f ∧ g)^(-1)(A) = f^(-1)(A) ∪ f^(-1)(B), then I know how to prove the rest, can anyone help me prove this part please? Thank you!

this is standard gluing lemma. – wanderer Mar 19 '14 at 05:47 — wanderer, Mar 19 '14 at 05:47

score 0 · Accepted Answer · answered Mar 19 '14 at 05:53

0

Let $x\in$ f^g$^{-1}(C)$ then f^g$(x)\in C$ and hence either $f(x)$ or $g(x)\in C$ which implies that $x\in f^{-1}(C)\cup g^{-1}(C)$. Proceed similarly for other side of inclusion.

Also, please write your question clearly...you've written "if I show f^g$^{-1}(A)$ = $f^{-1}(A)$ $\cup f^{-1}(B)$."

answered Mar 19 '14 at 05:53

wanderer

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I don't know how to use superscript here, I tried to write it in the right form but could not figure out how – JustAsk Mar 19 '14 at 06:05
superscript is not a problem here...you should've written $f^{-1}(A)\cup \bf{g^{-1}(A)}$ in RHS – wanderer Mar 19 '14 at 06:10

metric space, continuity, open and close

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