# Questions tagged [separation-axioms]

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27
questions

13
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2
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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 ...

4
votes

1
answer

165
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### 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
vote

1
answer

80
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### 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 ...

8
votes

0
answers

165
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### The pro-discrete space of quasicomponents of a topological space

Let $X$ be a topological space.
Consider the functor $P^X : \textbf{Set} \to \textbf{Set}$ that sends each set $Y$ to the set of continuous maps $X \to Y$.
It is not hard to check that $P^X : \textbf{...

3
votes

2
answers

170
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### Property ${\bf B}$ for families of large sets with small intersection

Let $\kappa\geq \aleph_0$ be a cardinal. If $X\neq \emptyset$ is a set, we say that a family ${\cal C}\subseteq {\cal P}(X)$ has property ${\bf B}$ if there is $S\subseteq X$ such that for all $C\in {\...

6
votes

1
answer

123
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### 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 ...

1
vote

1
answer

150
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### 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 ...

2
votes

1
answer

250
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### An example of a $T_1$ space where all closed $G_\delta$ sets are zero-sets, but it isn't normal

In Engelking's General topology, in the exercises section, there is Ju. M. Smirnov's characterization of normal spaces:
A $T_1$ space is normal iff the following properties hold (both):
Every closed $...

4
votes

1
answer

243
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### O. Frink's characterization of completely regular spaces

Def: Suppose X is topological space and B is a base for it. We say, that B is normal base, if following properties hold:
a. For any x∈X and A∈B, with x∈A, there exist A′∈B, such that x∉A′ and A∪A′=X.
...

5
votes

1
answer

171
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### Countable open covering of normal space

I read the following claim in Z.Frolik's article "A generalization of realcompact spaces" on page 135.
Two subset $M$ and $N$ of a space $X$ are called completely seperated if there exists a ...

6
votes

2
answers

355
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### Stone-Čech boundary is not extremally disconnected

Recall that a topological space is called extremally disconnected if the closure of every open subset is still open. Every discrete space is of course extremally disconnected, and the standard non-...

3
votes

0
answers

255
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### If the normalization is affine, is it affine? (if quasiaffine)

I was surprised to find out that, even if the normalization $X^\nu$ of a scheme $X$ is affine, $X$ may not be affine (remove the line $x=y$ from their example to make the source affine). In the ...

1
vote

0
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151
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### G Theory Localization Sequence without "quasiseparated"

Let $U \subseteq X$ be an open and $Z := X \setminus U$ its closed complement. I want a sequence
$$G_0(Z) \to G_0(X) \to G_0(U) \to 0.$$
However $X, U$ are not quasiseparated and perhaps not even ...

2
votes

1
answer

189
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### Are there minimal topological conditions on a space 𝑋 for it to have a countable separating set?

Are there minimal topological conditions on a space $X$ for it to have a countable separating set?
A separating set here is a set $D \subset C(X)$ (where $C(X)$ is the space of continuous functions ...

6
votes

3
answers

946
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### Is the lexicographic ordering on the unit square perfectly normal?

It's known that order topology is completely normal, so the lexicographic ordering on the unit square is also completely normal. It's also known that the lexicographic ordering on the unit square is ...

0
votes

1
answer

280
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### Mysior's example of not completely Hausdorff space

https://www.ams.org/journals/proc/1981-081-04/S0002-9939-1981-0601748-4/S0002-9939-1981-0601748-4.pdf
In this link, there is the example of regular space, that is not completely regular. This space ...

1
vote

1
answer

103
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### Separation in $l^1$ (Kreps-YanTheorem)

I have a question about the hypotheses of the Kreps-Yan Separation Theorem. I use the notation $l^p_+$ for the subspace of vectors all of whose coordinates are non-negative and define $l^p_- = -l^p_+$....

4
votes

1
answer

320
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### Supporting Hyperplane Theorem in Lp Spaces

Take C to be a closed convex set in $l^p$ (the space of sequences equipped with the $p$-norm where p>1) such that:
i) 0 is in C
ii) C is strictly larger than 0
iii) $C \cap -C =\{0\}$
iv) $C \...

4
votes

1
answer

259
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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 $$\{...

1
vote

1
answer

373
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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 ...

6
votes

1
answer

471
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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, ...

7
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0
answers

142
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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_\...

2
votes

1
answer

95
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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 $N\setminus O$.
...

2
votes

1
answer

473
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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 ...

3
votes

1
answer

411
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### Compact subsets and Hausdorffness of topology

We know that all closed subsets of a compact topological space $(X,\tau)$ are compact. If we add the Hausdorff condition on the topology $\tau$ we can see the equivalence of these conditions on a ...

3
votes

2
answers

682
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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 ...

8
votes

1
answer

1k
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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, ...