Let $t$ be the collection of all open sets of a topology on a set $U$ and let $a$ be a continuous function $U\to U$.
Can we prove the following for every set $E\subseteq U$:
$$a\left[\bigcap \left\{ D \in t \mid E \subseteq D \right\}\right] \subseteq \bigcap \left\{ D \in t \mid a [E] \subseteq D \right\}?$$
(Shorter: $a[\operatorname{cl}E]\subseteq\operatorname{cl}a[E]$.)
Or are there counterexamples?
Here $a[X]$ denotes the image of a set $X$ by function $a$.
Suppose that $z\in cl (E) $ and $f(z)\notin cl (a(E))$ hen there exists an open neighbourhood $U$ of $a(z)$ such that $U\cap cl (a(E))=\emptyset$ but this implies that $a^{-1} (U) \cap E\subset a^{-1} (U)\cap a^{-1} (cl (a(E))=\emptyset$ thus $z\notin cl (E) .$ Contradiction.