# 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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### In a Topological Vector Space T0 implies T3½ (completely regular)? And other separation properties.

I will describe my doubt. I know that in a TVS T1 implies T2. Now since a TVS admits a uniformisable topology, we have that T2 implies the uniform structure is separating. Now a separating uniform ...
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### Non-orientable 1-dimensional (non-hausdorff) manifold

Is there any nice example of a 1-dimensional non-hausdorff manifold that is not oriented? I have tried the line with two origins, but maybe something more exotic is needed?
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### Embedding of topological spaces into polytopes: complete regularity and metrizability.

In Dugundji's book Topology an interesting way to study topological spaces shows up frequerently: embedding them into polytopes which are defined by the author as arbitrary cartesian products of unit ...
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### “internal” definition of complete regularity?

There is something strange (I think) about the complete regularity separation axiom. Consider the definitions. T0 means for every two distinct points there is an open set containing exactly one of ...
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### Separability and the Nagata-Smirnov Metrisation Theorem

Definitions: Let $X$ denote a topological space throughout. If all singleton subsets of $X$ are closed, then we call $X$ Fréchet. If, given any closed subset $C \subset X$ and any point $x \in X - C$...
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### Separating closed sets in Moore plane / Niemytzki plane (Topology)

I spent the last few days trying to solve this exercise with little success, so I really hope someone here might be able to assist: Denote Moore plane by $M$, the $x$-axis by $R$ and the upper ...
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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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### Is $L^{n}$ normal, where $L$ denotes the closed long ray?

1.I am trying to prove that $L^{n}$, the $n$-$th$ product of closed long ray is normal, so that I can apply Tietze extension theorem to its closed subset and prove something else. I think I am able to ...
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### Is $\mathbb{R}^\omega$ endowed with the box topology completely normal (or hereditarily normal)?

Just out of curiosity, I'd like to know more properties of box topology. I found Is $\mathbb{R}^\omega$ a completely normal space, in the box topology? quite interesting, but unfortunately, it hasn't ...
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### I need a $T_D$ topological space that is not sober

in the book Frames and Locales by Jorge Picardo are defined two types of spaces: Sober spaces where the only meet-irreducible open sets are those in the form $X\setminus x^-$, where $x^-$ is the ...
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### Cofinite\discrete subspace of a T1 space?

Let $(X,\tau)$ be a $T_1$-space and $X$ is an infinite set. Then $(X,\tau)$ has a subspace homeomorphic to $(\mathbb{N},\tau_2)$, where $\tau_2$ is either the finite-closed topology or the discrete ...
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### Showing $\mathbb{N}$ with the topology generated from arithmetic progression is $T_2$ but not $T_3$

I'm trying to show that the natural numbers $\mathbb{N}=\{1,2,...\}$, with the topology that generated from the base $\{ (a+nb)_{n=0}^{\infty} | a,b\in \mathbb{N} ,gcd(a,b)=1\}$ is $T_2$ and not $T_3$...
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### Inverse image of singleton sets under continuous map between compact Hausdorff topological spaces

Let $X,Y$ be non-empty compact and Hausdorff topological spaces and $f:X \to Y$ be a continuous map. Take an element $y \in Y$. Question: Is $f^{-1}(\{y\})$ closed in $X$? Approaches and Ideas (...
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### Equivalence of $T_1$ axiom definition

I already know the following definition for $T_1$ axiom: $(1)$ Let $X$ a topological space. We say $X$ satisties $T_{1}$ axiom if all the finite subset of $X$ are closed. Now, I want to prove that ...
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### $(X,\mathscr T)$ is normal and each closed subsets of $X$ is a $G_{\delta}$ set.Then, $(X,\mathscr T)$ perfectly normal.

Prove the following result without using Urysohn's lemma. $(X,\mathscr T)$ is normal and every closed subset of $X$ is a $G_{\delta}$ set.Then, $(X,\mathscr T)$ perfectly normal. My effort: I ...
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### Every order topology is regular (proof check)

My proof: Let $X$ be an space with the order topology, $x \in X$ and $F$ a closed set that does not contain $x$. Then, the set $X-F$ is an open set that contains $x$, hence there is an open set (...
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### On exercise 14H from Willard's 'General Topology' book

$\newcommand{\R}{\mathop{\mathbf R}} \newcommand{\FN}{\mathop{\mathfrak N}}$ The exercise is in p. 99 of the book. It says the following: Let $X$ be a topological space, and let $B(X,\R)$ denote ...
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### Is $\mathbb R^J$ normal in the box topology when $J$ is uncountable?

Question: Is $\mathbb R^J$ normal in the box topology when $J$ is uncountable? I know $\mathbb R^J$ is not normal in the product topology, see "Proof" that $\mathbb{R}^J$ is not normal ...
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### Normality is not hereditary

For $(S,\tau_{disc})$ denote $A(S) = S \cup \{\infty_S\}$ as its onepoint compactification. Let $S,T$ be discrete spaces. Then clearly $X := A(S) \times A(T)$ is compact Hausdorff since each onepoint ...
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### Give an example of non-normal subspace of a normal space.

We know that a closed subspace of normal space is normal. My question was: why should other subspaces not work and then I came up with a counterexample. It is peculiar that any subspace of regular ...
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### Let X be a T1 space, and show that X is normal if and only if each neighbourhood of a closed set F contains the closure of some neighbourhood of F

Let $X$ be a $T_1$-space, and show that $X$ is normal if and only if each neighbourhood of a closed set $F$ contains the closure of some neighbourhood of $F$. I saw this statement in the book ...
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### Is every completely regular topology induced by some proximity?

A proximity space is a set endowed with a relation defining a notion of when two subsets are near or far apart. A proximity space induces a topology, and such a topology is always completely regular. ...
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### Topological groups are completely regular

I am studying topological groups, and I have been able to do quite a lot on my own by proving the propositions in this link on my own, but when I read up wikipedia that topological groups are all ...
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### $T_0$-space iff for each pair of $a$ and $b$ distinct members of X, $\overline{\{a\}}\neq\overline{\{b\}}$

Let $(X,\mathscr T)$ be topological space. Prove that $(X,\mathscr T)$ is $T_0$-space iff for each pair of $a$ and $b$ distinct members of X, $\overline{\{a\}}\neq \overline{\{b\}}.$ My attempt:-...
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### Is a completely regular space whose convergent sequences are eventually constant discrete?

If $X$ is a metrizable topological space where the only convergent sequences are eventually constant sequences, then $X$ must be a discrete space. But I'm interested in whether something stronger is ...
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### How do I prove $U (A_1)\cap V (A_2) \neq \emptyset$? Can You help me to find where do we arrive contradiction?

Let $(X,\mathscr T)$ be topological space with dense subset $D$ and a closed,relatively discrete subset $C$ such that $\mathscr{P}(D)\precsim$ $C.$ Then $(X,\mathscr T)$ is not normal. ...
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### Proof that a locally convex space $E$ is regular

Where a topological space $E$ is a regular space if, given any closed set $F$ and any point $x$ that does not belong to $F$, there exists a neighbourhood $U$ of $x$ and a neighbourhood $V$ of $F$ that ...
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### Prove that the function $f:X\to I=[0,1]$ by $f(x)=\min\{\frac{d(x,C)}{d(p,C)},1\}$ is continuous on $X$.

Question1. I can prove that Every metric space is Hausdorff. Where do they use this fact in this proof? Question2. Let $C$ be a closed subset of $X$ and $p\in X\setminus C$. How do I prove the ...
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### Regular $T_2$ space which is not completely regular.

Theorem 10. of Pontryagin's Topological Groups says that: Every Hausdorff topological group is completely regular. But is there exists a Regular $T_2$ space which is not completely regular?
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### Why are these two definitions of a perfectly normal space equivalent?

I've been skimming through some topology textbooks recently. Some sources, (such as Munkres' Topology and Willard's General Topology) define a space $(X,\mathcal{T})$ to be perfectly normal iff $X$ is ...
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