# Questions tagged [separation-axioms]

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13 votes
2 answers
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### Smooth Urysohn's lemma on Fréchet spaces

Let $V$ be a Fréchet topological vector space. Let $K_0$ and $K_1$ be two closed subsets which are disjoint. I wish to show the existence of a Fréchet-smooth function $f:V\to [0,1]$ whose restriction ...
• 42.5k
4 votes
1 answer
165 views

### A "simple" space with closed retracts but non-unique sequential limits

This question asking how KC ("Kompacts are Closed") and RC ("Retracts are Closed") are distinct has some good discussion, including a now-published example by Banakh and Stelmakh ...
• 1,045
1 vote
1 answer
80 views

### Reference for k-Hausdorff (in terms of compact T2 images)

In Rezk - Compactly generated spaces a k-Hausdorff property is defined, between weakly Hausdorff and unique sequential limits. On the other hand, a stronger notion of k-Hausdorff between $T_2$ and ...
• 1,045
8 votes
0 answers
165 views

6 votes
1 answer
123 views

### For which $X$ is $X\times I$ collectionwise normal?

Many normality-type properties can be characterised in terms of products with the unit interval $I=[0,1]$. For instance, if $X$ is a Hausdorff space, then; $X$ is normal and countably paracompact if ...
• 5,016
1 vote
1 answer
150 views

### Spaces whose interiors of retracts is a base of the topology

Definition:   topological space $\ X\$ is   r-basic $\ \Leftarrow:\Rightarrow\$ the interiors of retracts of $\ X\$ form a topological base of $\ X.$ Main question: Are r-basic spaces mentioned in ...
• 6,484
2 votes
1 answer
250 views

• 188
4 votes
1 answer
259 views

### 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 \{...
1 vote
1 answer
373 views

### 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 ...
• 225
6 votes
1 answer
471 views

### 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, ...
7 votes
0 answers
142 views

• 531
8 votes
1 answer
1k views

### 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, ...
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