Property of a function $f:X\to\mathcal{P}(X)$ defined on a topological space $(X,\tau)$: $x\in\overline{f(y)}\Leftrightarrow y\in\overline{f(x)}$

Question

I encountered the following interesting property of a function $f:X\to\mathcal{P}(X)$ defined on a topological space $(X,\tau)$:

$$x\in\overline{f(y)}\Leftrightarrow y\in\overline{f(x)}$$

where $\overline z$ is the closure of $z$. Does anybody know whether this property has a name or is equivalent to something else?

Answer 1

I'm sorry this question has nothing todo with the topology: if we define $g(x)=\overline{f(x)}$, the question reduces to the name of the property $x\in g(y)\Leftrightarrow y\in g(x)$. This is precisely when $g$ is symmetric when considered as a relation (every $g:X\to\mathcal{P}(X)$ can be considered as a $g\subseteq\mathcal(P)(X\times X)$).