Questions tagged [separation-axioms]

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2
votes
1answer
102 views

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
1answer
166 views

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
1answer
150 views

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 ...
3
votes
1answer
126 views

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
0answers
161 views

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
0answers
119 views

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
1answer
121 views

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
3answers
535 views

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
1answer
163 views

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
1answer
72 views

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
1answer
202 views

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
1answer
246 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
1answer
202 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 ...
6
votes
1answer
423 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
0answers
126 views

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
1answer
85 views

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
1answer
458 views

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
2answers
607 views

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
1answer
888 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, ...