metric space, continuity, open and close

Question

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!

Answer 1

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$^{-1}(A)$ = $f^{-1}(A)$ $\cup f^{-1}(B)$."