The separation-axioms tag has no usage guidance.

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### Generalizing the $T_0$-axiom

The starting point of this question is a slight reformulation of the $T_0$ separation axiom: A topological space $(X,\tau)$ is $T_0$ if for all $x\neq y\in X$ there is a set $U\in \tau$ such that $$\{...

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### Does regular $G_\delta$ imply normal?

I'm trying to prove that if every closed set in a topological space is regular $G_\delta$, then the space is normal. By regular $G_\delta$, I mean for any closed set $A$,
there exists a countable ...

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### Compact Lie group action on non-Hausdorff (but CGWH) space with Hausdorff quotient

Assume that we are in the following situation: a compact Lie group $G$ acts on a compact space $X$ which is not necessarily Hausdorff. $X$ is assumed to be compactly generated and weakly Hausdorff, ...

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### Is each Lindelof closed $\bar G_\delta$-set of a Tychonoff space functionally closed?

A subset $F$ if a topological space $X$ is called functionally closed if $F=f^{-1}(0)$ for some continuous map $f:X\to[0,1]$.
It is clear that each functionally closed set $F$ in $X$ is a closed $G_\...

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### Separating Differences of Open Sets

Has anyone ever considered something like the following separation axiom?
$(*)$ For any pair of open sets $O$ and $N$, there exist disjoint open sets containing $O\setminus N$ and $N\setminus O$.
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### Browder's fixed point theorem in non-Hausdorff topological vector spaces

Browder proved the following fixed point theorem in his 1968 Mathematische Annelen paper (Theorem 1):
Theorem. Let $K$ be a non-empty compact convex subset of a topological vector space $E$ (where we ...

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### Separation axioms

Reading about separation axioms, I wonder:
Is there a separation axiom weaker than $T_2$ but stronger than $T_1$? $T_{1.5}$?
I suppose there are some separation axioms stronger that $T_6$, how many ...

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### Stack with affine stabilizers but not quasi-affine diagonal

Give an example of a stack X with affine stabilizer groups and separated but not quasi-affine diagonal.
Remarks:
1) If X has finite stabilizer groups then the diagonal is quasi-finite and separated, ...