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2 in response to question edit

Note: this is a response to an earlier version of the question, and so is rather speculative. I am not sure that I can edit this to be much use in response to the newer version, so I'm leaving it as it stands for now, except to strikethrough a closing remark which was wide of the mark.

I think that the original questioner may have something like this in mind (though since I'm not a telepath, Jim, corrections/comments are welcome).

When we explain the definition of Hausdorff, we talk about separating points from other points using open neighbourhoods. Now if your intuition comes from metric spaces, these neighbourhoods are balls. So one might think that Hausdorffness is to do with separating things by balls, Furthermore, one might be thinking of some notion of minimal ball size -- this isn't what Hausdorffness is about, but bear with me! I'm trying to recreate a train of thought, not recapitulate the correct definitions -- and so get the idea that in "contexts which are Hausdorff" certain postulated objects -- the collection of all Widgets that satisfy the Sveshnikov-Pelikan equation - are forced to be finite because of "the need to separate constituent parts with balls". (See the original post's 4th para.)

Pursuing this train of thought, one might then wonder that if this postulated object is not finite, this is something to do with the failure of Hausdorffness. Again, I think this comes from a misapprehension about the Hausdorff separation condition; but at least this interpretation makes some sense of the original post's 1st para.

For the record: of course, things are more likely to be Hausdorff when you have more open sets, and of course a completely discrete space is Hausdorff for the trivial reason that every set is open. At the other extreme, a space with the indiscrete topology and more than two points has no chance of being Hausdorff. However, this has absolutely nothing to do with constraining the underlying set to be finite or infinite, contra the apparent guess of the original question. (If it helps: in the definition of Hausdorff, we don't constrain "the size of our open balls" before picking our two points; given any two points in, say, a metric space, we then have the freedom to choose a mesh size which will distinguish between them.) In particular, the sentence which starts "The topology of the multiverse would therefore be non-Hausdorff..." is in my view based on a misunderstanding, either of the word "Hausdorff" or the word "therefore"...

A comment for Ian, if I may: the reaction you got was because you started with a misuse of mathematical terminology, inserted into a question with words like "context" which are by nature philosophical/cultural rather than mathematical. It's that clash of tones which I think confused/irked some of the commenters.

1

I think that the original questioner may have something like this in mind (though since I'm not a telepath, Jim, corrections/comments are welcome).

When we explain the definition of Hausdorff, we talk about separating points from other points using open neighbourhoods. Now if your intuition comes from metric spaces, these neighbourhoods are balls. So one might think that Hausdorffness is to do with separating things by balls, Furthermore, one might be thinking of some notion of minimal ball size -- this isn't what Hausdorffness is about, but bear with me! I'm trying to recreate a train of thought, not recapitulate the correct definitions -- and so get the idea that in "contexts which are Hausdorff" certain postulated objects -- the collection of all Widgets that satisfy the Sveshnikov-Pelikan equation - are forced to be finite because of "the need to separate constituent parts with balls". (See the original post's 4th para.)

Pursuing this train of thought, one might then wonder that if this postulated object is not finite, this is something to do with the failure of Hausdorffness. Again, I think this comes from a misapprehension about the Hausdorff separation condition; but at least this interpretation makes some sense of the original post's 1st para.

For the record: of course, things are more likely to be Hausdorff when you have more open sets, and of course a completely discrete space is Hausdorff for the trivial reason that every set is open. At the other extreme, a space with the indiscrete topology and more than two points has no chance of being Hausdorff. However, this has absolutely nothing to do with constraining the underlying set to be finite or infinite, contra the apparent guess of the original question. (If it helps: in the definition of Hausdorff, we don't constrain "the size of our open balls" before picking our two points; given any two points in, say, a metric space, we then have the freedom to choose a mesh size which will distinguish between them.) In particular, the sentence which starts "The topology of the multiverse would therefore be non-Hausdorff..." is in my view based on a misunderstanding, either of the word "Hausdorff" or the word "therefore"...

A comment for Ian, if I may: the reaction you got was because you started with a misuse of mathematical terminology, inserted into a question with words like "context" which are by nature philosophical/cultural rather than mathematical. It's that clash of tones which I think confused/irked some of the commenters.