0
$\begingroup$

I am familiar with totally disconnected spaces. All components are singletons there.

Is there some terminology for spaces that satisfy the (stronger) condition of having only quasi-components that are singletons?

I thought of a name myself (totally separated) but maybe this is in use already for something else.

Also I am willing to subject to what is already there.

Thank you.

$\endgroup$
5
  • 1
    $\begingroup$ Apparently the term is "totally separated": proofwiki.org/wiki/Definition:Totally_Separated_Space $\endgroup$
    – Qiaochu Yuan
    Dec 5, 2020 at 9:17
  • $\begingroup$ @QiaochuYuan Thank you. Maybe once (long time ago) I knew and puzzling by search for a proper name it emerged while I was unaware. $\endgroup$
    – drhab
    Dec 5, 2020 at 9:56
  • 1
    $\begingroup$ In Engelking, General topology, a space is called hereditarily disconnected, if its components are singletons, hence what you call totally disconnected. In the historical notes to 6.2 he mentions that "at present the term totally disconnected is usually applied to a space X such that the quasi-component of any point $x \in X$ consists of the point x alone". Hence, according to Engelking, the term you are looking for is totally disconnected, indeed. $\endgroup$
    – Ulli
    Dec 6, 2020 at 18:20
  • $\begingroup$ @Ulli Thank you for this information. I have made my choice (totally separated) but it is a good thing to have notion of these diversities in definitions. Actually it would be better if they did not exist but that will always remain to be not more than wishful thinking. $\endgroup$
    – drhab
    Dec 6, 2020 at 18:56
  • $\begingroup$ @drhab Sure! I also don't like the term hereditarily disconnected, since only the empty space is really hereditarily disconnected. I prefer your settings as well. $\endgroup$
    – Ulli
    Dec 7, 2020 at 18:10

1 Answer 1

Reset to default
1
$\begingroup$

I believe "totally separated" actually is the standard term already - see e.g. here or here. Note that the latter source defines "totally separated" as "for all distinct points $p,q$, there is a partition of the space into open sets $U,V$ with $p\in U$ and $q\in V$," but that's clearly equivalent to all quasicomponents being singletons.

$\endgroup$
1
  • $\begingroup$ Thank you. Maybe once (long time ago) I knew and puzzling by search for a proper name it emerged while I was unaware. At the moment I am busy with Stone spaces (in my journey through logics). $\endgroup$
    – drhab
    Dec 5, 2020 at 9:57

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .