I am looking for the name and notation of the following separation axiom , temporarily denoted by $T\_i$ (where $i=\sqrt{-1}$ is the imaginary unit):


***Axiom $T\_i$***: For any point $x$ of a topological space $X$ and any neighborhood $O\_x$ of $x$ there is a closed subset $F$ in $X$ that contains $x$ and is contained in $O\_x$.


It is easy to see that a topological space $X$ satisfies the axiom $T\_i$ if and only if *each open set in $X$ is a union of closed sets*.


It is easy to check that the separation axiom $T\_1$ is equivalent to $T\_0+T\_i$.


The connected doubleton is an example of a $T\_0$-space which is not $T\_i$.
Any anti-discrete space satisfies $T\_i$ but not $T\_0$. So, the axioms $T\_0$
and $T\_i$ are incomparable.


**Question**: Is the axiom $T\_i$ known? If yes, where is it introduced and how is it denoted and called?



New separation axiom?
According to the [Wikipedia article about ${\mathrm T}\_1$ spaces](https://en.wikipedia.org/wiki/T1_space) your ${\mathrm T}\_i$-spaces are called $\it symmetric$ or ${\mathrm R}\_0$-spaces. There are several equivalent conditions, my personal favorite being that point closures are antidiscrete.


Unfortunately I was not able to pin down the initial place where this axiom has been introduced or used. The article refers to two books, but I could not find anything about ${\mathrm R}\_0$ there.


