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.