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.

2
votes
2answers
516 views

Prove that the space $\Bbb R_K$ is not regular.

Prove that the space $\Bbb R_K$ is not regular. where the basic open sets on $\Bbb R_K$ is given by $\{(a,b):a,b\in \Bbb R\}\cup \{(a,b)-K\}$ where $K=\{\dfrac{1}{n}:n\in \Bbb Z_+\}$. [Hint: ...
5
votes
1answer
141 views

A set very different from itself

Let $(X,\tau)$ be a regular space (having at least two points). Let's call $X$ self-different if the only homeomorphism $\phi:X\to X$ is the identity function. I know that you can have examples when $...
3
votes
1answer
74 views

Is every locally Banach, Hausdorff space regular?

I am working on some infinite dimensional differential geometry. I have tried proving a somewhat weaker statement than the above by replacing locally Banach with locally metrizable. But after some ...
2
votes
1answer
327 views

When a subspace of normal spaces is normal

It is well known that a closed subspace of a normal space is normal. I am looking for a condition $*$, such that the following statement is true. A subspace of a normal space is normal if and only if ...
2
votes
1answer
32 views

Can a point and a compact set in a Tychonoff space be separated by a continuous function into an arbitrary finite dimension Lie group?

Given a topological space $X$ which is Tychonoff (i.e., completely regular and Hausdorff), we know that given a compact set $K\subseteq X$ and a point $p \in X$ with $p\not\in K$, we can construct a ...
1
vote
1answer
121 views

A confusion regarding proof of “Every regular second countable space is normal”, given in Munkres

Let $X$ a topological space that is second countable and regular. Let $C,D$ be closed disjoint subspaces of $X$. By regularity, we can find for each $c \in C$ , disjoint open sets $U_c$ and $V_c$ such ...
0
votes
1answer
92 views

Linearly ordered X is regular

Prove that every linearly ordered space X is regular. Can anyone please help me with this proof? I started with letting $x$ belong to $x$ and take a nbhd $U=(a,b)$ of $x$ and then taking $A=(a,x)$, ...
0
votes
1answer
67 views

Separation of a set

Definition: Let $E \subseteq \Bbb R^n: U,V \subseteq \Bbb R^n $. $(U,V)$ is said to separate $E$ iff $ U,V \neq \emptyset$ $ U,V =$ relatively open in $E$ $ U \cap V = \emptyset$ $ U \cup V = E$ If $...
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)--...
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
0answers
134 views

Regularly open, co-zero sets in compact Hausdorff spaces

It follows from the definition of a completely regular space that such spaces have a base consisting of co-zero sets, that is, sets whose complement is the zero set of some real-valued, continuous ...
4
votes
0answers
79 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 ...
4
votes
0answers
221 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
0answers
150 views

Quotient space $N=\mathbb{R}^{2}/L$ is Hausdorff (or $T_1$) iff $L$ is closed

Let $L$ be a subset of $\mathbb{R}^{2}$ and let $N = \mathbb{R}^{2}/L$ be the quotient space obtained by identifying all points in $L$ to a single point. I need to prove that $N$ is Hausdorff $\...
4
votes
0answers
817 views

Is it true that every finite subset of a Hausdorff space has no limit point?

Let $X$ be a Hausdorff space. Assume a finite $A\subset X$ has a limit point $b$, say. Every pair $a\in A, b\in X$ has disjoint neighborhoods. Denote those neighborhoods of $b$ by $U_a$ for each $a\in ...
3
votes
0answers
216 views

If a space is regular and every point has a compact neighborhood, is it locally compact?

I attempted to find sufficient conditions for a space to be locally compact given that every point has a compact neighborhood, and I found that being regular is sufficient. I'm not sure my proof is ...
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 ...
2
votes
0answers
41 views

Topology under separation and countability

A nonempty product of spaces is $T_0$ if and only if each factor space is $T_{0}$. This is my solution: If $X_\alpha$ is a $T_0$ space for each $\alpha$ that belongs to $A$ and $x$ is not equal to ...
2
votes
0answers
394 views

All Borel $\sigma$-algebras are separating and countably generated?

I am studying the book Large Networks and Graph Limits by L. Lovasz. However, because of my CS background, I am not very knowledgable in topology and measure theory (but I try to catch up). In ...
2
votes
0answers
146 views

Intiution behind the theorem about locally compact Hausdorff spaces

I came across a theorem in Munkres' Topology. Theorem. Let $X$ be a Hausdorff space $X$ is locally compact iff given $x \in X$ and given a neighbourhood $U$ of $x$ there is a neighbourhood $V$ of $...
2
votes
0answers
233 views

Question on infinite T4 topological space

I have this question here from a course in general topology course I could not answer all of so I am asking here, it reads: Let $ (X,\tau) $ be a $ T_4 $ topological space (normal Hausdorff) with $ ...
2
votes
0answers
45 views

Set cannot be Hausdorff or compact with a non-subspace topology

$\mathcal{T}$ is the standard topology and $\mathcal{T}'$ is any other different topology, both on the unit interval $[0, 1]$. Then if $\mathcal{T}' \subsetneq \mathcal{T}$, $[0,1]$ with $\mathcal{T}'$...
2
votes
0answers
455 views

Why (or when) is the direct limit of compact spaces paracompact?

I'm working through Milnor and Stasheff's Characteristic Classes and got stuck in chapter 5, p.66, where some (supposedly) easy facts about paracompact spaces are assembled. One of these is: ...
2
votes
0answers
33 views

T1 axiom in dual space

My books talks about the conjugated space, but does it mean dual space? Are not the same thing? I don't understand why in the dual space $E^\ast$ of $E$, the separation axiom $T_1$ is satisfied and ...
2
votes
0answers
180 views

Alternative proof to Urysohn's lemma using $d(x,A)$.

Is there an alternative proof to Urysohn's lemma, that makes use of $d(x, A)$? Urysohn's lemma is: given a normal topological space $X$, for any disjoint closed sets $A$, $B$, there exists a ...
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 ...
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}\...
1
vote
0answers
27 views

Canonical example of non-Hausdorff Ellis group

Given a dynamical system (that is, for me, a group $G$ acting by homeomorphisms on a compact Hausdorff space $X$), the Ellis group (which can be defined in terms of the universal minimal flow (as here)...
1
vote
0answers
70 views

Separating closed sets by clopen sets in $\omega_1$.

I'm trying to verify that given two disjoint closed subsets of $\omega_1$ there is a clopen set $C$ containing one and disjoint from the other. I'm not seeing it at the moment, thanks for any help.
1
vote
0answers
750 views

Prove that every simply ordered set is a Hausdorff space in the order topology. The product of two Hausdorff space is again a Hausdorff space

In the book of General Topology by Munkres, at page 100, it is asked to prove that Every simply ordered set is a Hausdorff space in the order topology.The product of two Hausdorff space is again ...
1
vote
0answers
60 views

A sufficient condition for complete regularity?

If $X$ is a completely regular space, then the zero-set-neighborhoods of points form a neighborhood basis. Does the converse hold for Hausdorff spaces?I don't guess so.
1
vote
0answers
32 views

Theorem $10$ part $ii$ of the paper $I$ and $I^*$ convergence in topological spaces by P.Das and D.K. Lahiri

this paper pg 159-160 th $10.$ In the proof where $A=\{a_1,a_2...\}$ a countable set with $\bar A=F$ and the sequence $\{x_n\}$ is taken such that $x_n=a_i$ when $n\in M_i.$ Upto this point is fine. ...
1
vote
0answers
100 views

Separation axioms and when does there exist a connected space all whose proper connected subspaces are homeomorphic to the whole space?

This question is in the spirit of this question Does every non-singleton connected metric space $X$ contains a connected subset (with more than one point) which is not homeomorphic with $X$? ; ...
1
vote
0answers
138 views

Disjoint Union of Completely Regular Spaces

I am trying a new approach to an already-solved problem, but I need help to see if I'm on point. Munkres Chapter 53, question 6 [abridged] asks, given a covering map $p: E \to B$: Show that "if $B$ ...
1
vote
0answers
50 views

Is imaged of a Polish Hausdorff space under an injecitve map always Hausdorff?

I have a question about Hausdorff topological space. Question: Let $X,Y$ be topological spaces. If $X$ is a Polish space (i.e. $X$ is a separable and completely metrizable space.) and $Y=f(X)$ ...
1
vote
0answers
41 views

How to show $\mathbb{R^2}$ is sequentially connected without path-connectedness

Definitions: Connected: Not separated Separated: If $C$ is a subset of a metric space, then $(A, B)$ is a separation of $C$ if $C = A \cup B$, $A \neq \varnothing$, $B \neq \varnothing$, and ...
1
vote
0answers
39 views

How to find a hyperplane

Let $A, B ⊂ \mathbb{R}^n$ be two nonempty sets such that $A ∩ B = ∅$. $H(A, B) := \{(w, d) ∈ \mathbb{R}^{n+1} : \sup_{x\in A} \langle w,x\rangle ≤ d ≤ \inf_{y \in B} \langle w, y\rangle \}$ How do I ...
1
vote
0answers
110 views

Exercise in Section 2.4 of Singer & Thorpe

I'm trying to solve the exercise in Section 2.4 of Singer & Thorpe, which is to prove that if $S$ is a compact Hausdorff topological space and $(U_n)_{n \in \Bbb N}$ be a family of dense open sets,...
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
0answers
17 views

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?
0
votes
0answers
153 views

applications of topological spaces that are $T_1$, but are not $T_2$

If a topological space is $T_1$, its specialization preorder is the equality (so it is trivial). If a topological space is $T_2$ (Hausdorff), a limit of a proper filter is unique. I suppose this is ...
0
votes
0answers
337 views

Finding Interior and Derived set of some set

Let $X$ be the set of real numbers. Let a topology be $T_r$ generated by $\{(a,\infty): a$ is a real number$\}$ I have to find the derived set of $(0,1)$ in this topology. My answer is $X$, as any ...
0
votes
0answers
118 views

Closure of range of a compact operator is separable

Let $T \in K(\mathbb{H})$ , where $ K(\mathbb{H})$ is the space of all compact operators on Hilbert space $\mathbb{H}$. I need to show that closure of ${Range (T)}$ is separable. Any help is ...
0
votes
0answers
114 views

Examples of Toronto spaces

I remember $\dagger$ the notion of cardinality (a bijection), and the definitions of homeomorphism and Hausdorff space (separation axiom $T_2$). I would like to know if there exists examples of ...
0
votes
0answers
331 views

Proof of closed, continuous, surjective image of a *normal* space is normal.

Let $$p:X\rightarrow Y$$ be a closed, continuous, surjective map. I need to show that if $X$ is normal, then $Y$ is also normal. So I used this result: A space $X$ is normal iff for any closed ...
0
votes
0answers
157 views

Set-theoretic intersection of affine open subschemes.

Let $X$ be a separated scheme, $U,V \subseteq X$ open affine subschemes, $\Delta \colon X \to X \times X$ the diagonal morphism and $\pi_1, \pi_2 \colon X \times X \to X $ the natural projections, so ...