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Is there a special name for a topological space where the condition that every open set is clopen holds?

user729424
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groupoid
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    Do you have an example of this space, other than a discrete space? – John Dvorak Mar 06 '20 at 19:11
  • "totally disconnected" I think – Qurultay Mar 06 '20 at 19:12
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    @Qurultay No: Cantor space is totally disconnected but every co-singleton is open but not closed. – Noah Schweber Mar 06 '20 at 19:14
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    @JohnDvorak E.g. $X = {a, b, c, d, e}$, $\tau = { \emptyset, {a}, {b, c, d, e}, {b, c}, {a, d, e}, {d, e}, {a, b, c}, {a, b, c, d, e} }$. Note that the complement of every open set is also open. – groupoid Mar 06 '20 at 19:16
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    @JohnDvorak They're basically partitions. – Noah Schweber Mar 06 '20 at 19:16
  • Their topology is called partition topology. Probably you can extend the moniker to the space itself. –  Mar 06 '20 at 19:26
  • "clopen" ... for closed and open. Why don't you recall the definition of this word (besides, ugly and pedantic) I had never encountered it (though I have done a lot of topology). – Jean Marie Mar 06 '20 at 19:34
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    @JeanMarie 'clopen' is widely used, since at least the 60s. You must have had a limited exposure to topology. Maybe you did topology without reading much other people. In any case, be humble and don't use your knowledge or lack of it as a measure for what other people are used to do. The amount of literature is so vast that any single person easily ignores terminology that are widely known. –  Mar 06 '20 at 19:47
  • All right, consider I have had a limited exposure to topology in english language at that time. I try to imagine the same thing in French : "ferouvert" (lovely), in German "geschloffen" (very pretty) :) – Jean Marie Mar 06 '20 at 20:50
  • @JeanMarie "opgesloten" in Dutch has been heard by puristically minded topologists. – Henno Brandsma Mar 06 '20 at 23:22
  • @Henno Brandsma I surrender ! :) – Jean Marie Mar 07 '20 at 05:54
  • @JeanMarie French has ouvert-fermé and German uses abgeschlossene . – Mark S. Mar 07 '20 at 11:43
  • I guess the Stone topology can be an appropriate name to some extent. A non-trivial example which I remembered when I saw the post, is the natural topology on the space of complete types in a structure in Model Theory. Over there, model theorists refer to sets that are clopen and use the Stone topology term. – Maryam Ajorlou Mar 10 '20 at 18:05

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These spaces are quite rare, and I don't know a name for them. Indeed, they're really just partitions in disguise and consequently their $T_0$ifications are discrete, so they aren't really "topological" in flavor.

It's worth noting that it's not enough to have a base of clopen sets (= zero-dimensional with respect to the small inductive dimension); for example, Cantor space has a base of clopen sets but every co-singleton is open but not closed.

Noah Schweber
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