# Questions tagged [separation-axioms]

Separation axioms are properties of topological space which, roughly speaking, say in what way two points, a point and a closed set, or two closed sets can be "separated". Most important are $T_0$-spaces, $T_1$-spaces, Hausdorff, regular, completely regular and normal spaces.

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### Prove that the space $\Bbb R_K$ is not regular.

Prove that the space $\Bbb R_K$ is not regular. where the basic open sets on $\Bbb R_K$ is given by $\{(a,b):a,b\in \Bbb R\}\cup \{(a,b)-K\}$ where $K=\{\dfrac{1}{n}:n\in \Bbb Z_+\}$. [Hint: ...
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### A few standard results (on metrizability and relative separation strength) without choice?

I've been going back over some results from Munkres's Topology, and I'm curious about some things.... I know that Choice principles have some connection to the separation axioms (in ZF, at least)--...
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### If a space is regular and every point has a compact neighborhood, is it locally compact?

I attempted to find sufficient conditions for a space to be locally compact given that every point has a compact neighborhood, and I found that being regular is sufficient. I'm not sure my proof is ...
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### Bourbaki General Topology I: Exercise 20, sec. 8 ch.1

I have added a picture with the complete exercise. I'm interested only in c), I think I have proved a) and b). In c) we are given a topological space $X_0$ which is semi-regular (i.e. there exists a ...
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### Topology under separation and countability

A nonempty product of spaces is $T_0$ if and only if each factor space is $T_{0}$. This is my solution: If $X_\alpha$ is a $T_0$ space for each $\alpha$ that belongs to $A$ and $x$ is not equal to ...
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### All Borel $\sigma$-algebras are separating and countably generated?

I am studying the book Large Networks and Graph Limits by L. Lovasz. However, because of my CS background, I am not very knowledgable in topology and measure theory (but I try to catch up). In ...
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### Set cannot be Hausdorff or compact with a non-subspace topology

$\mathcal{T}$ is the standard topology and $\mathcal{T}'$ is any other different topology, both on the unit interval $[0, 1]$. Then if $\mathcal{T}' \subsetneq \mathcal{T}$, $[0,1]$ with $\mathcal{T}'$...
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### Why (or when) is the direct limit of compact spaces paracompact?

I'm working through Milnor and Stasheff's Characteristic Classes and got stuck in chapter 5, p.66, where some (supposedly) easy facts about paracompact spaces are assembled. One of these is: ...
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### T1 axiom in dual space

My books talks about the conjugated space, but does it mean dual space? Are not the same thing? I don't understand why in the dual space $E^\ast$ of $E$, the separation axiom $T_1$ is satisfied and ...
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### Alternative proof to Urysohn's lemma using $d(x,A)$.

Is there an alternative proof to Urysohn's lemma, that makes use of $d(x, A)$? Urysohn's lemma is: given a normal topological space $X$, for any disjoint closed sets $A$, $B$, there exists a ...
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### Behavior of points and compact subsets of Hausdorff spaces

It is quite straightforwad to see that many prpoperties are shared by points and compact subspaces of Hausdorff topologies, for example in terms of separation properties. I was wondering if there is ...
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