Lemma 5.21.5. Let $f : X \to Y$ be a continuous map of topological spaces. Let $T \subset X$ be a subset. If $f$ is a homeomorphism of $X$ onto a closed subset of $Y$ and $T$ is nowhere dense in $X$, then also $f(T)$ is nowhere dense in $Y$.
Proof. Because $f(X)$ is closed in $Y$ and $f$ is a homeomorphism of $X$ onto $f(X)$, we see that the closure of $f(T)$ in $Y$ equals $f(\overline{T})$. Hence if $V \subset Y$ is open and contained in the closure of $f(T)$, then $U = f^{-1}(V)$ is open and contained in $\overline{T}$. Hence $U = \emptyset $, which in turn shows that $V = \emptyset $ as desired. $\square$
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