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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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Example 2, Sec. 31, in Munkres' TOPOLOGY, 2nd ed: Normality of $\mathbb{R}_l$ — Why are these two sets disjoint?

The set $\mathbb{R}$ of real numbers with the lower limit topology having as a basis the collection of all closed-open intervals $[a, b)$, where $a, b \in \mathbb{R}$ with $a < b$, is denoted by $\...
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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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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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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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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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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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Show that there exists a set $U$ which is both open and closed and $x \in U \subseteq V$.

Let $X$ be a compact topological space. Suppose that for any $x, y \in X$ with $x \neq y$, there exist open sets $U_x$ and $U_y$ containing $x$ and $y$, respectively, such that $$ U_x \cup U_y = X\...
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Quotient to make $X$ a $T_1$ space

Let $X$ be a topological space. We define a relation on $X$: $$x \approx y : \quad \Leftrightarrow \quad x \in \overline{\{y\}}.$$ In general $\approx$ is no equivalence relation since it lacks ...
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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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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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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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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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$(X,\mathscr T)$ 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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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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Singletons closed in infinite T1 Product Topology? [duplicate]

[Similar to this, but I think the question I'm asking isn't covered] This is part of a problem in section 9.2 of S. Morris's "Topology Without Tears". I'm confused about the following construction: ...
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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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Can we prove the theorem without injectivity of $f$?

Let $(X,\mathscr T)$ be normal space, let $(Y,\mathscr U)$ be topological spaces, and let $f:X\to Y$ be closed continuous function that maps $X$ onto $Y$. Then $(Y,\mathscr U)$ is normal. Proof. ...
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$X$ $T_3$ and $f:X\rightarrow Y$ cont. open and closed $\Longrightarrow$ $f(X)$ Hausdorff

The title corresponds to Theorem 14.6 of General topology, by Stephen Willard. Without loss of generality, we can assume that $f$ is onto too. Then, since $f$ is open, by a previous result it is ...
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Is every completely regular topology induced by some topological vector space?

Every topological vector space is completely regular. My question is, is the converse true? That is, is every completely regular topology induced by some topological vector space? If not, does ...
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What separation axiom is necessary for existence of neighborhood which closure is a subset of another given neighborhood

I'm looking for the weakest separation axiom, which gives the following property: Let $A$ be neighborhood of the point $x$. Then there exists another neighborhood $B$ of $x$, such $\overline{B}\...
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Another equivalent condition for being $T_1$−space $(X,\mathscr T)$ is normal.

A $T_1-$space $(X,\mathscr T)$ is normal iff for each pair of disjoint closed subsets $C$ and $D$ of $X$ there are open sets $U$ and $V$ such that $C\subseteq U$, $D\subseteq V,$ and $\overline{U}\...
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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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An equivalent condition for being $T_1-$space $(X,\mathscr T)$ is normal.

A $T_1-$space $(X,\mathscr T)$ is normal iff for each closed subset $C$ of $X$ and each open set $U$ such that $C\subseteq U$, there is an open set $V$ such that $C\subseteq V$ and $\overline{V}\...
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Nested interestection forms neighbourhood basis

Let $X$ be a topological space and $x \in X$. Suppose that there exists a countable collection $(U_n)_{n \geq 1}$ of open sets such that $U_{n + 1} \subseteq U_n$ and $\bigcap\limits_{n \geq 1} U_n = \...
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How do I prove that there exists a bijection from power set of $D$ to $C'\subseteq C.$

I was proving the Moore's Plane $(X,\mathscr T)$ is not normal. Using the theorem. Let $C=\{(x,y)\in X:y=0\}$ be a closed and relatively discrete subset of $X$. Let $D=\{(x,y)\in X:x\in \mathbb Q \...
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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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$T_0$-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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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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Compact group acting on regular space

Let $G$ be a compact topological group, $X$ be a regular topological space. Then the quotient space given by the continuos action of $G$ on $X$, $X/G$ is also regular. Here's my attempt, though I feel ...
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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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Locally compact Hausdorff then Tychonoff

This is my proof. If $X$ is locally compact Hausdorff then for each $x \in X$ its compact neighborhoods form a neighborhood basis at $x$. For an arbitrary closed set $C \not\ni x$ let $V \subset X \...
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Prime Integer Topology is $T_2$ but not $T_3$

According to this $\pi$-Base page, the "Prime Integer Topology" is an example of a topological space which is $T_2$ but not $T_3$. The space is defined as $(\mathbb{Z}^+,\tau)$ where $\tau$ is the ...
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Separating points in a compact Hausdorff space.

Suppose that $X$ is a compact Hausdorff space and let $\{x_1,\ldots,x_n\}$ be a finite collection of points in $X$. It it possible to find open neighbourhoods $U_i$ of $x_i$ such that such that $x_j \...
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What does it mean if you take the zero vector in the hyperplane separation theorem?

If you have two disjoint sets V and W, which are compact and convex, then by the separation theorem there exists a vector u such that uv< uw. However what does it mean if you take u=0?
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Does there exist a continuous function separating these two sets $A$ and $B$

True or False: There exists a continuous function $f : \mathbb{R}^2 → \mathbb{R} > $such that $f ≡ 1$ on the set $\{(x, y) \in \mathbb{R}^2 : x ^2+y^2 =3/2 \}$ and $f ≡ 0$ on the set $B∪\{(x, y)...
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For a closed $G_\delta$ set $F\subseteq X,$ does there exist a continuous function $f:X\to [0,1]$ such that $f=0$ on $F$ and $f\neq 0$ outside $F?$

(All spaces are Hausdorff.) This question is a variant of my previous question. Let $X$ be a completely regular space, that is, for every closed set $F\subseteq X$ and $x\not\in F,$ there exists a ...
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For every closed set $F$ in a completely regular space $X,$ does there exist a nonzero continuous function $f:X\to [0,1]$ such that $f=0$ outside $F?$ [closed]

Let $X$ be a completely regular space, that is, for every closed set $F\subseteq X$ and $x\not\in F,$ there exists a continuous function $g:X\to [0,1]$ such that $g(F) = \{0\}$ and $g(x) =1.$ ...
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A space is Hausdorff iff…

Here's an exercise from Marco Manetti's "Topologia" book (ex. 3.59 in the italian version) that i'm stuck on: Prove that a topological space $X$ is $\mathrm{T2}\iff\{x\}=\bigcap\limits_{U\in\,\...
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Can every locally compact Hausdorff space be recognized as a subspace of a cube that has an open underlying set?

In this question cubes are topological spaces of the form $[0,1]^J$ with product topology and $[0,1]$ with usual topology. Further a space is a Tychonoff space if and only if it is a completely ...
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For the existence of one-point compactification, do we need locally compactness?

In the book Topology by Munkres, at page 184, it is given the existence and uniqueness of one point compactification of a locally compact Hausdorff space; however, in the existence part, I can't see ...
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What topological spaces satisfy yet another property involving relatively compact sets?

This is a follow-up to my questions here and here. A subset of a topological space is called relatively compact if its closure is compact. Let's call a sequence $(U_n)$ of open sets a bounding ...
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What topological spaces satisfy another property involving relatively compact sets?

This is a follow-up to my question here. A subset of a topological space is called relatively compact if its closure is compact. My question is, what kind of topological spaces satisfy the following ...
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What topological spaces satisfy a property involving relatively compact sets?

A subset of a topological space is called relatively compact if its closure is compact. My question is, what kind of topological spaces satisfy the following property: for every relatively compact ...
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Contradiction in the proof a space with every compact subspace is not Hausdorff

Let $(X,\tau)$ be an infinite topological space with the property that every subspace is compact. Prove that $(X,\tau)$ is not a Hausdorff space. A space is Husdorff if for all $a,b\in X$ then there ...
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Prove or disprove if a quotient map from X to Y with Y Hausdorff, then X is Hausdorff. [closed]

For two open disjoint subsets U and V, I want to show their pre-images are disjoint open subsets of X or not. But I have no idea how to do it. Any help would be appreciated. Thanks in advance!
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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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Proving a certain topology to be non-Hausdorff

I'm doing this exercise and I don't know how to attack the second part: Let $\tau$ be a topology on $\mathbb{N}$, with $\tau=\{\emptyset\}\cup\{\{0,1,2,...,n\}|n\in\mathbb{N}\}\cup\{\mathbb{N}\}$. ...
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Proof verification: All metric spaces fulfill the “Hausdorff property”.

I am currently studying an introductory book to complex analysis. Within the first chaper the complex numbers are presented in various ways aswell as their structure regarding to different points of ...