# continuous on metric space

• Nov 14th 2011, 06:17 PM
wopashui
continuous on metric space
Let $f: (M,d)\to (N,p)$. Prove that f is continuous if and only if $f(\text{cl}(A))\subset \text{cl}(f(A))$ for every $A \subset M$ if and only if $f^{-1}(\text{int}(B))\subset \text{int}(f^{-1}(B))$ for every $B \subset N$.
Give an example of a continuous f such that $f(\text{cl}(A)) \ne \text{cl}(f(A))$ for some $A \subset M$.

I have shown that the closure case is true, i need some help on the interior case and the example.
I was told to use the fact that inverse image of every open set is open
• Nov 15th 2011, 04:18 AM
Plato
Re: continuous on metric space
Quote:

Originally Posted by wopashui
Let $f: (M,d)\to (N,p)$. Prove that f is continuous if and only if $f^{-1}(\text{int}(B))\subset \text{int}(f^{-1}(B))$ for every $B \subset N$.

Note that if $O$ is an open set in $N$ then $\text{int}(O)=O$.
If $f(x_0)\in O$ then $x_0\in (f^{-1}(O))=f^{-1}(\text{int}(O))\subseteq \text{int}(f^{-1}(O))$