Let $X$ be a metric space, and $Y\subset X$ a subset. A point $x\in X$ is adherent to $Y$ if $B(x;r) \cap Y \neq \emptyset$ $\forall r > 0.$ The closure of $Y$ is then defined as $\bar Y := \{x\in X \mid x \text{ is adherent to } Y\}$.
I am trying to prove the next statement. Is there any mistake or error?
$f$ is a homeomorphism. Show that $f(\bar A) \subset \overline{f(A)}$.
Since $f$ is continuous $x\in \bar A \implies f(x)\in f(\bar A)$. (Is this right? or trivial?) I want to show $f(x)\in f(\bar A) \implies f(x)\in \overline{f(A)}$. Consider any open ball $V = B(f(x);r)$. Since $f$ is continuous, $f^{-1}(V)$ is an open subset of $X$ and $x\in f^{-1}(V)$.
Because $x\in \bar A$, there exists an element in the intersection of $A$ and any open set. i.e. $f^{-1}(V)\cap A \neq \emptyset$. Take $y\in f^{-1}(V)\cap A$, then
$$ f(y)\in f(f^{-1}(V)\cap A) \subset V\cap f(A).$$
From this, we see that $V\cap f(A)\neq \emptyset$ for any open ball $V=B(f(x);r)$. By the definition of closure, $f(x)\in \overline{f(A)}$.
Best Answer
The proof is fine. I just want to point a few things:
"Since $f$ is continuous $x\in \bar A \implies f(x)\in f(\bar A)$.
(Is this right? or trivial?)" Recall the definition $f(\overline{A}) =\{f(x) \in Y \mid x \in \overline{A}\}$, so this step is actually trivial.It's overkill asking that $f$ is a homeomorphism. We only need $f$ to be continuous.
Assuming $f$ continuous, the reciprocal is actually true, too.
Do notice that you didn't really used the fact that $V$ was an open ball. This, in fact, is not important - the argument holds with $V$ being any open set containing $x$.