Let $f:X \to Y$ be a continuous map between topological spaces and $A \subset X$. Is it true that $\overline {f( \overline
A)}= \overline {f(A)}$?
[Math] Closure of continuous image of closure
general-topology
general-topology
Let $f:X \to Y$ be a continuous map between topological spaces and $A \subset X$. Is it true that $\overline {f( \overline
A)}= \overline {f(A)}$?
Best Answer
Yes. Recall that $f$ is continuous if $f[\overline{S}]\subseteq\overline{f[S]}$ for all $S\subseteq X$. Applying this, we get that $f[\overline{A}]\subseteq\overline{f[A]}$. Since the set on the right is closed we also get $\overline{f[\overline{A}]}\subseteq\overline{f[A]}$. For the other inclusion, simply note that $f[A]\subseteq f[\overline{A}]$ and take closures.