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?

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    $\begingroup$ Something like $T_i$ is necessary for defining e.g. the pseudocharacter of a topological space. The pseudocharacter of a point $x$ in a topological space $X$ is the smallest cardinality $|\mathcal U|$ of a family $\mathcal U$ of neighborhoods of $x$ such that $\{x\}=\bigcap\mathcal U$. It is well-defined if and only if the open set $X\setminus\{x\}$ is a union of closed sets. Usually in this case topologists requre the axiom $T_1$ but we see that the weaker $T_i$ suffices. $\endgroup$ – Taras Banakh Apr 16 '16 at 8:38
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    $\begingroup$ Also if one like to consider networks of closed sets (s)he will need the axiom $T_i$. $\endgroup$ – Taras Banakh Apr 16 '16 at 8:43
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    $\begingroup$ To define the Borel chierarchy it is useful to have that every open set is a countable union of closed sets. This is a countable version of the axiom $T_i$. $\endgroup$ – Taras Banakh Apr 16 '16 at 8:47
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    $\begingroup$ Wikipedia calls this $R_0$ or symmetric. $\endgroup$ – მამუკა ჯიბლაძე Apr 16 '16 at 10:20
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    $\begingroup$ მამუკა ჯიბლაძე, maybe write this your answer as the standard answer, which I will accept and will close the question as answered. $\endgroup$ – Taras Banakh Apr 16 '16 at 10:41

According to the Wikipedia article about ${\mathrm T}_1$ spaces 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.

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    $\begingroup$ Both the nLab article ncatlab.org/nlab/show/separation+axioms and the Wikipedia article were largely written by Toby Bartels, so I'll ask him if he can verify the references. $\endgroup$ – Todd Trimble Apr 16 '16 at 23:48
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    $\begingroup$ Ah: Toby wrote this Wikipedia article: en.wikipedia.org/wiki/Separation_axiom and there he gives Schechter as another reference. Maybe the $R_i$ axioms are there... $\endgroup$ – Todd Trimble Apr 17 '16 at 0:03
  • $\begingroup$ @Todd Thanks for the highly relevant reference! I'll still leave the one I had, for the sake of the list of equivalent conditions for ${\mathrm R}_0$. Looked in Schechter, could not find anything outside ${\mathrm T}_0$ there either. $\endgroup$ – მამუკა ჯიბლაძე Apr 17 '16 at 5:01
  • $\begingroup$ It is not easy to google for these, what I stumbled upon is a paper where they refer to a 1978 book "General topology" by Császár and to "Quasi-Uniform Topological Spaces" by Murdeshwar & Naimpally (1966). Unfortunately I don't have access to either one of these. $\endgroup$ – მამუკა ჯიბლაძე Apr 17 '16 at 5:11
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    $\begingroup$ I guess the term symmetric comes from the fact that these spaces are exactly those for which: $x \in \mathrm{cl}\{y\} \Leftrightarrow y \in \mathrm{cl}\{x\}$. $\endgroup$ – Ramiro de la Vega Apr 25 '16 at 14:02

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