# continuous on metric space

• Nov 14th 2011, 05:17 PM
wopashui
continuous on metric space
Let $\displaystyle f: (M,d)\to (N,p)$. Prove that f is continuous if and only if $\displaystyle f(\text{cl}(A))\subset \text{cl}(f(A))$ for every $\displaystyle A \subset M$ if and only if $\displaystyle f^{-1}(\text{int}(B))\subset \text{int}(f^{-1}(B))$ for every $\displaystyle B \subset N$.
Give an example of a continuous f such that $\displaystyle f(\text{cl}(A)) \ne \text{cl}(f(A))$ for some $\displaystyle A \subset M$.

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, 03:18 AM
Plato
Re: continuous on metric space
Quote:

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

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