Definition talk:Hausdorff Space/Definition 1

Separated by Open Sets

The definition separated by open sets does not yet exist, but I think adding the page could lead to confusion with Definition:Separated Sets and Definition:Separated Points which don't require $U$ and $V$ to be disjoint.

Therefore I suggest that the statement:

That is:

$\left({S, \tau}\right)$ is a $T_2$ space if and only if every two elements in $S$ are separated by open sets.

be dropped from the definition. --Leigh.Samphier (talk) 01:08, 28 March 2020 (EDT)

Not sure I agree. What we do is we make sure we are careful to define exactly what we mean by "separated" in all the situations where the word is used. --prime mover (talk) 02:07, 28 March 2020 (EDT)

Fair enough. I've found Definition:Separated by Neighborhoods which is an equivalent definition to what is required for Definition:Separated by Open Sets. So I'll add a second definition to Definition:Separated by Neighborhoods, add an equivalence page and then create the page Definition:Separated by Open Sets as a redirect. --Leigh.Samphier (talk) 03:51, 28 March 2020 (EDT)

Are we sure that Definition:Separated by Neighborhoods is the same thing as Definition:Separated by Open Sets? A neighbourhood is not necessarily an open set unless it is an open neighbourhood. There are subtleties here which we have glossed over before which have caused mistakes to appear on $\mathsf{Pr} \infty \mathsf{fWiki}$. I had a discussion with this with Lord_Farin on this subject a few years ago. --prime mover (talk) 04:03, 28 March 2020 (EDT)

Absolutely sure. An open set is a neighbourhood, so Separated by Open Set implies Separated by Neighborhoods. And a Neighbourhood contains an open set that contains the set/point, so Separated by Neighbourhoods implies Separated by Open Sets. --Leigh.Samphier (talk) 04:14, 28 March 2020 (EDT)

But if a T4 set is characterised as being separated by neighbourhoods, and a T2 set is characterised as being separated by open sets -- then that means a T2 set and a T4 set are the same thing, which they are not. Seriously, I think there's something more subtle going on here. --prime mover (talk) 04:28, 28 March 2020 (EDT)

EDIT: got it, T2 is "points" separated by open sets, and T4 is "disjoint closed subsets" separated by neighbourhoods. Okay then, but I still wonder why the terminology is different between the two in Steen and Seebach. --prime mover (talk) 04:30, 28 March 2020 (EDT)

... and now I've just reread the note on Definition:Hausdorff Space#Equivalence of Definitions which comments on the difference in terminology. Yes I know, I wrote that, it was a long time ago :-) --prime mover (talk) 04:33, 28 March 2020 (EDT)