1
$\begingroup$

When people ask general topology questions asking for a proof or a counterexample to a claim, people often do not mention whether they want a counterexample which satisfies higher separation axioms (complete regularity and above) or whether a counterexample which does not satisfy higher separation axioms would be sufficient.

However, general topologists, functional analysts, and other mathematicians find spaces which satisfy higher separation axioms to be more natural than spaces that only satisfy lower separation axioms (in part because converging to multiple points is strange). On the other hand, order theorists and algebraic geometers use spaces which only satisfy lower separation axioms for their own reasons.

Should we encourage users to specify whether they prefer or require the counterexamples to satisfy higher separation axioms? If so, then how should we go about encouraging users to consider whether they would prefer the counterexamples to satisfy higher separation axioms? Should we think about including tags for higher separation axioms and for lower separation axioms or are there already too many tags to begin with?

If I see a question on MathOverflow asking for a topology counterexample that has been answered with a non-Hausdorff counterexample, then I would search for a completely regular counterexample since a completely regular counterexample is often more natural, and if I find a completely regular counterexample, I would post that counterexample as an answer to the question.

I think it is a good practice for general topology answerers to seek to give counterexamples which satisfy higher separation axioms rather than lower separation axioms.

$\endgroup$
  • $\begingroup$ The AMS mathematics subject classification draws the line between lower separation axioms and higher separation axioms at complete regularity. I agree with the AMS classification that complete regularity (instead of Hausdorff) is the correct point at which to draw the line between lower separation axioms and higher separation axioms. I cannot think of very many naturally occuring topological spaces which are Hausdorff but not completely regular though. $\endgroup$ – Joseph Van Name Jun 19 '16 at 5:20
  • 4
    $\begingroup$ I’d say that regardless of what the original poster cares for, some users will surely be interested in higher separation axioms. So, an example satisfying more separation axioms is always useful additional information and you should post it. $\endgroup$ – Emil Jeřábek Jun 19 '16 at 12:39
  • $\begingroup$ I never saw a problem with omnipresent points (read: dense singletons). It's not weird. If anything, it's weird that we're making a big deal out of this. I mean, everything infinite is inherently weird. (Sorry, I had to vent... :P) $\endgroup$ – Asaf Karagila Jun 19 '16 at 22:24
  • $\begingroup$ I have no problem with spaces that are not Hausdorff either since there are many very nice topological spaces which are not Hausdorff. That being said, saying property A+regular$\rightarrow$property B is vastly different than saying property A$\rightarrow$ property B for all topological spaces. I would even venture to say that completely regular spaces are much more similar to completely regular frames (frames are point-free topological spaces) than completely regular spaces are to non-Hausdorff spaces. $\endgroup$ – Joseph Van Name Jun 20 '16 at 1:51
  • $\begingroup$ Do you often find there to be any ambiguity? I usually find I can tell from the flavor of the question whether it's asked from the point of view of analysis / algebraic geometry / order theory / etc. If it's unclear, surely it's easy enough to add a comment like "I assume you want the space to be Hausdorff?" $\endgroup$ – Nate Eldredge Jun 26 '16 at 2:50
  • $\begingroup$ I find this ambiguity often enough for me to comment on it especially for questions of the form "Does there exist a space X that satisfies this special property" where the asker is asking the question out of curiosity. $\endgroup$ – Joseph Van Name Jun 28 '16 at 5:57

You must log in to answer this question.

Browse other questions tagged .