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.

91
votes
3answers
4k views

Does this property characterize a space as Hausdorff?

As a result of this question, I've been thinking about the following condition on a topological space $Y$: For every topological space $X$, $E\subseteq X$, and continuous maps $f,g\colon X\to Y$, ...
56
votes
4answers
21k views

$X$ is Hausdorff if and only if the diagonal of $X\times X$ is closed

Let $X$ be a topological space. The diagonal of $X \times X$ is the subset $$D = \{(x,x)\in X\times X\mid x \in X\}.$$ Show that $X$ is Hausdorff if and only if $D$ is closed in $X \times X$. First,...
33
votes
3answers
1k views

If every continuous $f:X\to X$ has $\text{Fix}(f)\subseteq X$ closed, must $X$ be Hausdorff?

Given a function $f:X\to X$, let $\text{Fix}(f)=\{x\in X\mid x=f(x)\}$. In a recent comment, I wondered whether $X$ is Hausdorff $\iff$ $\text{Fix}(f)\subseteq X$ is closed for every continuous $f:X\...
28
votes
5answers
959 views

Is the closure of a Hausdorff space, Hausdorff?

$(X,\mathcal T)$ is a topological space which has a dense Hausdorff subspace. Is $X$ Hausdorff?
27
votes
5answers
8k views

$X/{\sim}$ is Hausdorff if and only if $\sim$ is closed in $X \times X$

$X$ is a Hausdorff space and $\sim$ is an equivalence relation. If the quotient map is open, then $X/{\sim}$ is a Hausdorff space if and only if $\sim$ is a closed subset of the product space $X \...
25
votes
2answers
852 views

How many compact Hausdorff spaces are there of a given cardinality?

This is a question I found myself wondering about recently. I eventually figured out the answer myself, but as this doesn't seem to be written down anywhere easy to find on the Internet I decided to ...
21
votes
1answer
2k views

When is $C_0(X)$ separable?

Recall that a compact Hausdorff space is second countable if and only if the Banach space $C(X)$ of continuous functions on $X$ is separable. I'm looking for a similar criterion for locally compact ...
17
votes
3answers
774 views

Construction of a Hausdorff space from a topological space

Let $X$ be a topological space. Is there a Hausdorff space $HX$ and a continuous function $i:X\rightarrow HX$ such that for any Hausdorff space $A$ and a continuous function $j:X\rightarrow A$, there ...
16
votes
1answer
9k views

The product of Hausdorff spaces is Hausdorff

I'm confused how it can be true that the product of an infinite number of Hausdorff spaces $X_\alpha$ can be Hausdorff. If $\prod_{\alpha \in J} X_\alpha$ is a product space with product topology, ...
15
votes
4answers
1k views

Why are ordered spaces normal? [collecting proofs]

Greets This is a problem I wanted to solve for a long time, and finally did some days ago. So I want to ask people here at MSE to show as many different answers to this problem as possible. I will ...
15
votes
2answers
1k views

How big can a separable Hausdorff space be?

It is just an idea (might be wrong) but, i think that if a Hausdorff space, say $X$, contains too many elements, then a countable subset cannot be dense in it. Does there exist a cardinality that any ...
15
votes
6answers
7k views

Examples for subspace of a normal space which is not normal

Are there any simple examples of subspaces of a normal space which are not normal? I know closed subspace of a normal space is normal, but open subspace in most cases which I can think of are also ...
15
votes
4answers
1k views

Looking For a Neat Proof of the Fact that the Grassmannian Manifold is Hausdorff

$\newcommand{\R}{\mathbf R}$ Let $V$ be an $n$-dimensional vector space and $k$ be an integer less than $n$. A $k$-frame in $V$ is an injective linear map $T:\R^k\to V$. Let the set of all the $k$-...
15
votes
2answers
343 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 ...
14
votes
1answer
1k views

When is a quotient by closed equivalence relation Hausdorff

Let us say for an arbitrary topological space $X$ that it has property $\dagger$ if for any closed equivalence relation $\sim$ on $X$ (closed as a subset of $X^2$), the quotient space $X/{\sim}$ is ...
13
votes
2answers
1k views

Topology on $\mathbb{R}$ strictly coarser (resp. finer) than the usual one which is still Hausdorff (resp. connected)

The following are simple observations. Suppose $\mathcal{T}_1,\mathcal{T}_2$ are two topologies on a set $X$ such that $\mathcal{T}_1$ is finer than $\mathcal{T}_2$. If $( X ,\mathcal{T}_2 )...
13
votes
2answers
921 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 ...
13
votes
0answers
464 views

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)--...
10
votes
3answers
1k views

Example of Hausdorff space $X$ s.t. $C_b(X)$ does not separate points?

We know the Stone-Weierstrass theorem for locally compact Hausdorff spaces (LCH) which states the following: Theorem: Suppose $X$ is LCH. A subalgebra $\mathcal{A}$ of $C_0(X)$ is dense if and only ...
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 ...
9
votes
1answer
469 views

Cantor cubes are universal for totally disconnected compact Hausdorff spaces

Can anyone tell me how to show that a totally disconnected compact Hausdorff space is homeomorphic to a closed subspace of a product of discrete two-point spaces? I cannot think of a known example of ...
9
votes
2answers
629 views

What application is there for a non-Hausdorff topological space?

I'm learning basic topology and as I understand it, a good way to intuit what an open set is, is that it determines which elements are near each other. However, in a non-Hausdorff space, it would be ...
9
votes
2answers
151 views

The Hausdorff property versus closedness of the diagonal in the context of convergence spaces

Given a topological space $X$, the following are equivalent: Given points $x$ and $y$, there exist neighborhoods $A$ and $B$ of $x$ and $y$ respectively satisfying $A \cap B = \emptyset$. Every ...
8
votes
2answers
691 views

A non-Hausdorff space with a Hausdorff subspace [closed]

Can anyone give an example of a non-Hausdorff space that contains a Hausdorff subspace?
8
votes
3answers
419 views

A strong Hausdorff condition

Is the following strong form of Hausdorff equivalent to usual Hausdorff? $X$ is strong Hausdorff if given distinct elements $x,y$ in $X$ there are open sets $U,V \subseteq X$ with $x \subseteq U$, ...
8
votes
2answers
1k views

Why does this argument show that a retract of a Hausdorff space is closed?

This Question gives the following argument for why the retract of a Hausdorff space is closed: Proof. Let $x∉A$ and $a=r(x)∈A$. Since $X$ is Hausdorff, $x$ and $a$ have disjoint neighborhoods $U$...
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?
8
votes
4answers
2k views

A finite Hausdorff space is discrete

Theorem: $X$ is a finite Hausdorff. Show that the topology is discrete. My attempt: $X$ is Hausdorff then $T_2 \implies T_1$ Thus for any $x \in X$ we have $\{x\}$ is closed. Thus $X \setminus \{x\}$ ...
8
votes
1answer
1k views

Is every linear ordered set normal in its order topology?

I'm trying to prove (or disprove) that every linear ordered set $(X, <_X)$ is normal in its order topology. I was able to prove $(X,<_X)$ is hausdorff, simply by taking two open intervals with ...
8
votes
1answer
186 views

Is Arens Square a Urysohn space?

Example 80 in Steen-Seebach1 is called Arens Square. It is defined in the book as follows: Let $S$ be the set of rational lattice points in the interior of the unit square except those whose $x$-...
8
votes
1answer
460 views

Give an example of a non compact Hausdorff space such that $\Delta$ is closed but $Y$ is not Hausdorff

Suppose $X$ is compact Hausdorff space and $f : X \to Y$ be a quotient map. Then it is well known that $Y$ is Hausdorff iff $\Delta =\{(x,y) \mid f(x)=f(y) \}$ is closed in $X \times X$. For example ...
7
votes
2answers
3k views

If two continuous maps into a Hausdorff space agree on a dense subset, they are identically equal [duplicate]

Let $f, g : X \to Y$ be continuous functions. Assume that $Y$ is Hausdorff and that there exists a dense subset $D$ of $X$ such that $f(x) = g(x)$ for all $x \in D$. Prove that $f(x) = g(x)$ for all $...
7
votes
1answer
2k views

Is a metric space perfectly normal?

I typically like to practice my knowledge on a specific concept by doing proofs using one definition of a term, and then doing the same proofs using an equivalent definition (without inducing the ...
7
votes
1answer
1k views

Every infinite Hausdorff space has an infinite discrete subspace

I want to show that any infinite Hausdorff space contains an infinite discrete subspace. I am motivated by the role of $\mathbb N$ in $\mathbb R$. We know that if a Hausdorff space is finite, then it ...
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 ...
7
votes
1answer
68 views

A second opinion on a proof in topology

My friend and I were looking over some homework questions for an upcoming test in introductory topology, and one of the questions on the homework was to show that a metric space is normal. What we ...
7
votes
1answer
157 views

$T_{2.5}$ topology without coarser metric topology

Let $(X,\tau)$ be a second-countable $T_{2.5}$ space, where with $T_{2.5}$ I mean that any distinct points are separated by closed neighborhoods. Does there have to be some metrizable second-countable ...
7
votes
1answer
139 views

Separation axioms in uniform spaces

I have some problems understanding the proof of the following lemma: Lemma: Let $x \in X, \ \ \ U, W \in \mathcal{U}, \ \ \ \mathcal{T(U)}$ is the topology on $X$. If there exists $V \in \mathcal{U}$ ...
7
votes
1answer
235 views

Subsets $\Omega\subseteq X$ such that any continuous $f:\Omega\to Y$ can be extended continuously to $F:X\to Y$

This is a reference request. Below is a prelude to the main questions. Let $n$ be a positive integer. Characterize all subsets $\Omega$ of $\mathbb{R}^n$ such that any continuous function $f:\...
7
votes
0answers
272 views

Urysohn's Lemma needn't hold in the absence of choice. Alternate terminology for inequivalent definitions of “normal” spaces?

A topological space $\langle X,\tau\rangle$ is said to be normal if any two disjoint closed subsets are separated by open sets, meaning that for disjoint $E,F\subseteq X$ with $X\setminus E,X\setminus ...
6
votes
3answers
7k views

$f,g$ continuous from $X$ to $Y$. if they are agree on a dense set $A$ of $X$ then they agree on $X$

Problem: Suppose $f$ and $g$ are two continuous functions such that $f: X \to Y $ and $g : X \to Y $. $Y$ is a a Hausdorff space. Suppose $f(x) = g(x) $ for all $x \in A \subseteq X $ where $A$ ...
6
votes
4answers
3k views

Regular but not normal space

This was an exam question of ours: Let $\chi$ be the set, $\chi = \left \{ a, b, c, d \right \}$. Create a topology $\tau$ on $\chi$ so that $\left ( \chi ,\tau \right )$ is regular but not ...
6
votes
3answers
869 views

Why study non-T1 topological spaces?

I can understand (somewhat) why one would want to study non-Hausdorff topologies, since for example the Zariski topology is so important to algebraists, and the weak topology generated by lower ...
6
votes
2answers
673 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?
6
votes
1answer
2k views

Is the set of fixed points in a non-Hausdorff space always closed?

It is not hard to show that if $f: X \rightarrow X$ is a continuous map and $X$ is a Hausdorff space, then the set of fixed points is closed in $X$. We basically just look at the diagonal and consider ...
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 ...
6
votes
1answer
1k views

Question about quotient of a compact Hausdorff space

I am reading the book 'Algebraic Topology' by Tammo Tom Dieck. On page 12 in the proposition 1.4.4 he states that : Let $X$ be a compact Hausdorff space and $f : X \rightarrow Y$ be a quotient ...
6
votes
3answers
943 views

Compactness and Strictly Finer Topologies.

If $(A,\tau{_1})$ is a compact Hausdorff space and $\tau{_2}$ is a strictly finer topology on $X$, can $(A, \tau_{2})$ be compact?
6
votes
2answers
812 views

Prove: the countable product of regular topological spaces is regular.

Prove: the countable product of regular topological spaces is regular. Label the countable product of $X_i$ as $X$. Given $x \in X$ and $U$ a closed set s.t. $ x \notin U$, let's find disjoint ...
6
votes
2answers
168 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 ...