Symmetric closure in a topological space

Question

Consider the following property of a (typically not $T_1$) topological space $X$: given any $x_1,x_2\in X$, we have that $x_1\in \overline{\{x_2\}}$ if and only if $x_2\in \overline{\{x_1\}}$.

It's equivalent to saying that closures of points are minimal closed sets.

Does this property have a name?

Answer 1

This type of space is apparently called an $R_0$ space; this terminology is backed up by

• The Wikipedia page for Specialization (pre)order.
• The Wikipedia page for $T_1$ spaces.
• This article found by googling "$R_0$ space".