Do the two axioms that "distinct points/closed set and point outside are precisely separated by continuous function" have their names? (Compared to perfectly normal, maybe we should call them perfectly Hausdorff & perfectly regular?) What are the relationship between these two and other separation axioms?
2026-03-26 04:28:23.1774499303
Separation axioms about precisely separated by continuous function
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Such spaces are called "completely Hausdorff" and "completely regular", respectively. You can check that here and here. You can also find over there all the causal relationships between them and other separation axioms (or other topological properties).
A word of caution: I wouldn't assume anyone knows these names. I knew them just because I just checked them, and I assume I will momentarily forget them.
EDIT.
Apparently I misunderstood the question -- I guess "precisely separated" means that $f^{-1}(0)=\{x\},f^{-1}(1)=\{y\}$ rather than just $f(0)=x,f(1)=y$.
So I have a new answer: I would guess there is no name for these properties. Of course, it makes perfect sense to call them "perfect Hausdorffness" and "perfect regularity", but, if even Pi Base does not list them, I have to assume it's something no one really talks about. But perhaps someone will prove me wrong and give some references.