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.

Filter by
Sorted by
Tagged with
72
votes
4answers
26k 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
5answers
10k 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 \...
9
votes
3answers
9k 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 $A$ ...
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 ...
97
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 $f,g\colon X\to Y$, ...
6
votes
1answer
639 views

Mrówka spaces are first-countable

The construction of a Mrówka space (a $\Psi$-space) is not clear for me. Because of this I could not see why it is first-countable, locally compact, and Hausdorff. Coud you give me some help about ...
7
votes
1answer
2k 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 ...
6
votes
1answer
3k views

Non-T1 Space: Is the set of limit points closed?

I have shown that the set of limit points in $T_1$-space is closed, and the proof uses the $T_1$ axiom, so I was wondering: Given $X,$ not necessarily $T_1,$ and any $A\subset X,$ is it necessarily ...
18
votes
1answer
12k views

Quotient Space of Hausdorff space

Is it true that quotient space of a Hausdorff space is necessarily Hausdorff? In the book "Algebraic Curves and Riemann Surfaces", by Miranda, the author writes: "$\mathbb{}P^2$ can be viewed as the ...
8
votes
1answer
2k 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 ...
7
votes
1answer
934 views

A continuous bijection from a compact space to a $T_2$ space is always a homeomorphism

This problem is bugging me for some time. $f:(X,\mathcal{T}) \to (Y,\mathcal{T}')$ is a continuous bijection where $(X,\mathcal{T})$ is compact and $(Y,\mathcal{T}')$ is $T_2$ (i.e. Hausdorff and $...
3
votes
2answers
489 views

Is it true that every normal countable topological space is metrizable?

I've been reading about and working on various proofs about metrizabililty. I'm having trouble answering the following question: Is it true that every normal countable topological space is metrizable? ...
17
votes
4answers
909 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 ...
6
votes
1answer
3k views

Prove that a separable metric space is Lindelöf without proving it is second-countable

This a problem from my exam: Prove that a separable metric space is a Lindelöf space I know that a separable metric space is second-countable, and second-countable implies the Lindelöf property. ...
10
votes
4answers
3k 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 $X \setminus \{x\}$ ...
1
vote
2answers
173 views

GRE 9367 #62: Prove $X=[0,1]$ in lower limit topology ($[a,b)$) is not compact, is Hausdorff and is disconnected.

GRE9367 #62 Ian Coley's solution: Sean Sovine's solution: Prove $X$ is not compact. My first proof was similar to Ian Coley's, but I came up with another proof: If $X$ is compact, ...
1
vote
1answer
610 views

Show that any metrizable space $X$ is Hausdorff

I wish to show that any metrizable space $(X,\mathcal{T})$ is Hausdorff Proof attempt: Let $d$ be the metric that generates the topology on $X$. Pick two points $x,y \in X$, we wish to produce two ...
18
votes
1answer
12k 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, ...
9
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 ...
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 $...
6
votes
3answers
1k 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?
11
votes
1answer
1k 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 ...
8
votes
3answers
484 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 $x \subseteq U$, ...
1
vote
1answer
119 views

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 ...
6
votes
2answers
828 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?
5
votes
1answer
760 views

Help understanding proof of every topological group is regular

I found a proof that any topological group is regular here, but I got lost in the last part. The whole argument goes like this: Consider the map $f:G \times G \to G$ defined by $f(a,b)=ab^{-1}$. This ...
25
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 ...
15
votes
1answer
2k 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 ...
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 ...
17
votes
6answers
9k 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 ...
9
votes
2answers
742 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 ...
14
votes
3answers
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 $( X ,\mathcal{T}_2 )$ is ...
7
votes
3answers
497 views

Every topological space can be realized as the quotient of some Hausdorff space.

Prove that every topological space can be realized as the quotient of some Hausdorff space. I tried to show this by using the intersection of two open sets in $x$ (for $f:z\to x$).
2
votes
3answers
1k views

Examples of a quotient map not closed and quotient space not Hausdorff

Is there any example of a closed relation $\sim$ on a Hausdorff space $X$ such that $X/\sim$ is not Hausdorff? Also, is there any example of a closed relation ~ on a Hausdorff space $X$ such that a ...
14
votes
2answers
1k 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 ...
6
votes
3answers
1k 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 ...
5
votes
1answer
2k views

Show that a connected regular space having more than one point is uncountable

Two questions on which I am stuck: 1.Show that a connected normal space having more than one point is uncountable. 2.Show that a connected regular space having more than one point is uncountable. ...
5
votes
4answers
8k views

Show that a locally compact Hausdorff space is regular.

Show that a locally compact Hausdorff space $(X,\tau)$ is regular. I have already shown that a compact Hausdorff space is regular. My textbook proposes 2 methodes, but I get stuck at both. The first ...
3
votes
1answer
345 views

Normal space- equivalent condition: $\forall U, V$ - open $: U \cup V = X \ \ \ \exists F \subset U \ , G \subset V$ - closed $: F \cup G = X$

Could you help me prove that $X$ is normal $\iff$ for all open $U, V$such that $U \cup V = X$ there are closed $F \subset U,\ G \subset V,\ F \cup G = X?$ I've been trying to prove it using other ...
8
votes
5answers
672 views

Necessity of being Hausdorff in the definition of compactness?

According to R Engelking - General Topology: A topological space $X$ is called a compact space if $X$ is a Hausdorff space and every open cover of $X$ has a finite subcover, i.e., if for every ...
6
votes
3answers
2k views

Prove that $Y$ is Hausdorff iff $X$ is Hausdorff and $A$ is a closed subset of $X$

Let $X$ be a topological space and $A$ a subset of $X$. On $X\times\{0,1\}$ define the partition composed of the pairs $\{(a,0),(a,1)\}$ for $a\in A$, and of the singletons $\{(x,i)\}$ if $x\in X\...
8
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 ...
5
votes
1answer
3k views

$[0,1)\times[0,1)$ (lower limit topology) is a regular, but not a normal topological space

Let $X=[0,1)\times[0,1)$, $\tau$ its topology with base $$\beta = \{ [a,b)\times[c,d): 0 \leq a < b \leq 1, 0 \leq c < d \leq 1 \}\;.$$ Please help me prove, that it is regular, but not a ...
5
votes
1answer
998 views

Every minimal Hausdorff space is H-closed

A Hausdorff topological space $(X,\mathcal T)$ is called H-closed or absolutely closed if it is closed in any Hausdorff space, which contains $X$ as a subspace.$\newcommand{\ol}[1]{\overline{#1}}\...
3
votes
1answer
373 views

Is a covering space of a completely regular space also completely regular

I'm trying to solve a problem in Munkres' Topology book. Let $p: E \rightarrow B$ be a covering map and suppose that $B$ is completely regular (for any closed subset $A$ and disjoint point $a$ there ...
3
votes
1answer
2k views

Finer topologies on a compact Hausdorff space

If we have such topological space $(X,\mathcal{T})$ that it is compact and Hausdorff, then we can say that for any other topology $\mathcal{H}$ on $X$ such that $\mathcal{T}\subseteq\mathcal{H}$, the ...
5
votes
1answer
1k 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} =...
4
votes
0answers
238 views

Is the Alexandroff double circle compact and Hausdorff?

I recently encountered the Alexandroff double circle. The underlying set is $C = C_1 \cup C_2$, where $C_i$ is the circle of radius $i$ and centre $0$ in the complex plane. The basic open sets are: $\...
4
votes
1answer
157 views

Closed surjection that does not preserve regularity

Def Map $p\colon X\rightarrow Y$ is perfect if it is a closed surjection and $p^{-1}\left(\left\{y\right\}\right)$ is compact for each $y\in Y$ It is well known that perfect maps preserve regularity, ...
3
votes
2answers
590 views

$X$ second countable locally compact Hausdorff implies $C(X)$ separable?

In the post When is $C_0(X)$ separable? , it is argued that if $X$ is second countable and locally compact Hausdorff, then $C_{0}(X)$ is separable. Is it also true that $C(X)$ is separable under the ...