9
$\begingroup$

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?

$\endgroup$
10
  • 2
    $\begingroup$ 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. $\endgroup$ Apr 16, 2016 at 8:38
  • 1
    $\begingroup$ Also if one like to consider networks of closed sets (s)he will need the axiom $T_i$. $\endgroup$ Apr 16, 2016 at 8:43
  • 1
    $\begingroup$ 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 $T_i$. $\endgroup$ Apr 16, 2016 at 8:47
  • 6
    $\begingroup$ Wikipedia calls this $R_0$ or symmetric. $\endgroup$ Apr 16, 2016 at 10:20
  • 4
    $\begingroup$ მამუკა ჯიბლაძე, maybe write this your answer as the standard answer, which I will accept and will close the question as answered. $\endgroup$ Apr 16, 2016 at 10:41

1 Answer 1

12
$\begingroup$

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.

$\endgroup$
8
  • 2
    $\begingroup$ 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, 2016 at 23:48
  • 1
    $\begingroup$ 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, 2016 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. $\endgroup$ Apr 17, 2016 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. $\endgroup$ Apr 17, 2016 at 5:11
  • 2
    $\begingroup$ I guess the term symmetric comes from the fact that these spaces are exactly those for which: $x \in \mathrm{cl}\{y\} \Leftrightarrow y \in \mathrm{cl}\{x\}$. $\endgroup$ Apr 25, 2016 at 14:02

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.