Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

0
votes
2answers
31 views

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 ...
6
votes
2answers
670 views

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?
0
votes
1answer
25 views

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 ...
2
votes
2answers
138 views

“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 ...
5
votes
2answers
68 views

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 $x \in X - C$...
6
votes
3answers
223 views

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 ...
1
vote
0answers
42 views

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 ...
3
votes
2answers
67 views

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\...
2
votes
1answer
61 views

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 ...
1
vote
1answer
54 views

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 ...
2
votes
1answer
90 views

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 ...
0
votes
1answer
43 views

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 ...
6
votes
2answers
633 views

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 ...
2
votes
2answers
76 views

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 $T_3$...
1
vote
1answer
39 views

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 (...
0
votes
1answer
24 views

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 ...
3
votes
2answers
115 views

$(X,\mathscr T)$ 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 ...
4
votes
1answer
2k views

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 (...
0
votes
1answer
121 views

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 ...
4
votes
0answers
78 views

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 ...
5
votes
1answer
973 views

Product of $T_1$ spaces is $T_1$

I am trying to prove that the product of $T_1$ spaces is also $T_1$. Here is a proof, is it correct? $\{ X_i \}_{i \in I}$ are T1 $\Rightarrow$ $\prod_{i \in I} X_i$ is T1 Proof: Let $\bar{x} =...
0
votes
1answer
30 views

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: ...
2
votes
0answers
77 views

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 ...
1
vote
1answer
30 views

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. ...
1
vote
2answers
90 views

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 ...
15
votes
2answers
342 views

Do Hausdorff spaces that aren't completely regular appear in practice?

Completely regular spaces include all metrizable spaces, topological vector spaces, and topological groups in general. In fact, they are exactly the uniformizable spaces. Complete regularity is ...
2
votes
1answer
35 views

$X$ $T_3$ 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 ...
6
votes
2answers
167 views

quotient topology doesn't preserve separation axioms

According to Wikipedia Quotient topology is ill-behaved with respect to Separation Axioms,locally compactness and simply connectedness. I have examples to support this argument for locally ...
1
vote
3answers
1k views

First Countable Spaces are Hausdorff or Not?

Does first countable imply Hausdorff? If not, what is an example of first countable space that is not Hausdorff?
0
votes
0answers
24 views

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 ...
0
votes
1answer
61 views

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 ...
4
votes
1answer
34 views

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}\...
2
votes
1answer
857 views

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 ...
7
votes
3answers
1k views

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 ...
0
votes
1answer
360 views

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 ...
0
votes
1answer
17 views

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. ...
1
vote
1answer
23 views

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}\...
1
vote
0answers
19 views

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}\...
13
votes
2answers
919 views

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 ...
0
votes
1answer
17 views

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 = \...
0
votes
2answers
25 views

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 \...
0
votes
1answer
46 views

$T_0$-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:-...
2
votes
2answers
36 views

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 ...
1
vote
2answers
64 views

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. ...
3
votes
1answer
106 views

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 ...
0
votes
1answer
31 views

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 ...
8
votes
2answers
233 views

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?
10
votes
1answer
979 views

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 ...
0
votes
1answer
54 views

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 \...
3
votes
1answer
66 views

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 ...