Skip to main content

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
4 votes
0 answers
39 views

When must a space generated by compacts also be generated by Hausdorff compacts?

I'm interested in comparing $k_1$-spaces, spaces whose topologies are witnessed by their compact subspaces, and $k_3$-spaces, spaces whose topologies are witnessed by their compact and Hausdorff ...
Steven Clontz's user avatar
5 votes
0 answers
68 views

Is this relation making a space Hausdorff?

Given a topological space $X$ there are ways of introducing relations $\rho$ on it so that $X/\rho$ becomes a Hausdorff space. Here I will describe two relations which I will call inner and outer (...
cnikbesku's user avatar
  • 631
1 vote
0 answers
33 views

For compact Hausdorff spaces, is countable pseudocharacter equivalent to first countable? [duplicate]

Let $X$ be a compact $T_2$ space. Is $X$ first countable if, and only if, $X$ has countable pseudocharacter? Note: I have already proven that every $T_1$ first countable space has countable ...
Alman's user avatar
  • 11
3 votes
1 answer
51 views

Well-definedness of the projection associated to the sheaf of germs of a presheaf

I'm currently reading Izu Vaisman's Cohomology and differential forms ($1973$) having never studied sheaf theory before, so I will briefly write down the definitions in case they don't match with ...
Bruno B's user avatar
  • 5,912
1 vote
2 answers
43 views

In a regular space with a $\sigma$-locally finite network, is every closed set a $G_\delta$?

In a regular space with a $\sigma$-locally finite network, is every closed set a $G_\delta$? I'm inclined to think so based upon the basic argument of Theorem 2.8 here, but there are a few moving ...
Steven Clontz's user avatar
0 votes
1 answer
48 views

Understanding the Separation theorem

Separation theorem: Let $P, Q⊆\mathbb{R}^d$ be disjoint compact convex sets. Then there exist $v∈ \mathbb{R}^d$ and $c_1, c_2∈\mathbb{R}$ with $c_1<c_2$ such that $x.v≤c_1$ for every $x∈P$ $x.v≥...
D. S.'s user avatar
  • 313
1 vote
0 answers
47 views

Character of dense subspace of regular space

Define the character of a topological space $X$ at $x$ to be the minimal cardinality of a base at $x$, denoted by $\chi(x,X)$. Suppose $M$ is a dense subspace of a regular space $X$, prove that $\chi(...
BlowingWind's user avatar
3 votes
0 answers
73 views

Must locally compact and weakly Hausdorff spaces be regular?

In a recent pull request to the π-Base, it was observed that all locally compact and KC (Kompacts are Closed) spaces are regular: since compacts are closed, each local neighborhood base of compacts is ...
Steven Clontz's user avatar
1 vote
1 answer
37 views

How to prove this "cofinite topology with a distinguished point" is T3.5?

In Ryszard Engelking's General Topology, its Example 1.1.8 defines a topology as follows: Let $X$ be an arbitrary infinite set, $x_0$ a point in $X$ and $\mathcal{O}$ the family consisting of all ...
BlowingWind's user avatar
0 votes
1 answer
25 views

Let $(Y,T_{y})$ be a T1 and first countable and $(X,T_{x})$ as below , prove that a function between topological spaces X to Y is continuous iff

Sorry if its elementary, but here goes. Let X be uncountable and $x_{0}$ be an arbitrary point of X, $T_{x}$ is the topology generated by the following basis $\beta $ = {${x}$ such that $x \neq x_0$} $...
Victor Hugo's user avatar
0 votes
2 answers
67 views

Cannot understand how this topology is non Hausdorff

Consider the following topology over $\mathbb{R}$: $\tau = \{\emptyset, \mathbb{R}, \mathbb{R}\setminus [0, 3], [0, 3]\}.$ I cannot understand why this is non Hausdorff. I read here around that a ...
Heidegger's user avatar
  • 3,492
1 vote
1 answer
50 views

Two sets having strongly seperated points are strongly seperated themselves

Definition :- Two sets $A$ and $B$ are strongly seperated if there exist open neighbourhoods $U$ and $V$ such that $A \subseteq U , B \subseteq V , U\cap V=\phi$ Question :$A$ and $B$ are two compact ...
user-492177's user avatar
  • 2,589
6 votes
2 answers
119 views

If $(X,T)$ is a compact Haussdorff space and X is countable, then the set $\{x:\{x\} \in T\} $ is dense in $(X,T)$

If $(X,T)$ is a compact Haussdorff space and X is countable, then the set $\{x:\{x\} \in T\} $ is dense in $(X,T)$ ? Apologize if it is elementary. but here goes what I`ve tried so far Since that it ...
Victor Hugo's user avatar
7 votes
1 answer
152 views

Are Hausdorff countably compact topological groups always normal?

A colleague and I have a result that shows that for Hausdorff countably compact W-spaces, being a topological group implies normality. But it occurred to us that (being not super experienced working ...
Steven Clontz's user avatar
1 vote
1 answer
29 views

Singletons in regular spaces

I need help with a statement. Let $X_1$ and $X_2$ be topological spaces and consider $X = X_1 \times X_2$. Suppose the product space $X$ is regular, that is, for any closed subset $C \subset X$ and ...
Joel Marques's user avatar
1 vote
1 answer
50 views

If $X$ is regular and $A$ is a closed subset of $A$, then $X/A$ is Hausdorff [duplicate]

If $X$ is regular and $A$ is a closed subset of $A$, then $X/A$ is Hausdorff How can I use to canonical quotient map $q: X \to X/A$ to prove the result? I tried picking distinct elements $x, y \in X/...
pera erdir's user avatar
1 vote
0 answers
36 views

How can I directly prove that a characterization of perfectly normal using $G_\delta$ sets implies completely normal?

There are several questions on Math.SE showing how perfectly normal implies completely normal. However, I am attempting to prove this fact without appealing to a characterization that requires the ...
Steven Clontz's user avatar
3 votes
1 answer
42 views

What separation axioms are held by Gustin's sequence space?

Gustin's sequence space is defined as counterexample #125 in Steen and Seebach's Counterexamples in Topology. Notably, while most of its separation axioms are established in its General Reference ...
Steven Clontz's user avatar
0 votes
5 answers
333 views

Does a metric space need to be a Hausdorff space? [closed]

I need to prove that: Let $X$, $Y$ be metric compact spaces, $f:X \to Y$ be continuous and bijective. Then $f$ is an homeomorphism. But I have been investigating, and there's no such theorem, ...
Roberto Ruíz's user avatar
-1 votes
1 answer
73 views

How to prove that he looped line topology is Tychonoff? [closed]

I came to this question when I was solving a problem in Willard's General Topology (p. 36). I am new to topology and I want to prove that the looped line topology is Tychonoff space. My idea was to ...
Nova h's user avatar
  • 9
0 votes
0 answers
51 views

Strong Seperation between hypreplane and an affine subspace

Let $h \in \mathbb{R}^n\backslash \{0\}$ and $r \in \mathbb{R}$. A sure that $M$ is an affine subspace of $\mathbb{R}^n$ with $H(h, r) \cap M=\phi$. I tried to use Banach separation theorem to show ...
bruno's user avatar
  • 425
3 votes
1 answer
105 views

Is there a first countable, $T_1$, weakly Lindelof, sequentially compact space which is not also compact?

Spinning things out from a recent question, let's recall that every second countable sequentially compact space is compact, because every Lindelof countably compact space is compact. We cannot weaken ...
Steven Clontz's user avatar
6 votes
1 answer
116 views

Different Hausdorff topologies with the same continuous mappings

Fix a set $X$. For a topology $\tau$ on $X$, let $F_\tau = \{f: X \to X \mid f \text{ continuous w.r.t. } \tau\}$. What would be an example of a set $X$ and two Hausdorff topologies $\tau$ and $\sigma$...
Smiley1000's user avatar
  • 1,649
1 vote
1 answer
94 views

Necessity of Hausdorff-ness in "continuous function determined by its values on a dense subset"

It's well-known that if a continuous function taking values in a Hausdorff space is uniquely determined by its specification on a dense subset of the domain. Now, I contemplate on the necessity of ...
Atom's user avatar
  • 4,119
3 votes
1 answer
131 views

Fully normal implies paracompact without a $T_1$ assumption?

It's well-known that a $T_1$ topological space is fully normal if and only if it is $T_2$ and paracompact. It appears, looking at the proofs from Henno Brandsma's nice exposition here and here, that ...
M W's user avatar
  • 9,941
2 votes
1 answer
61 views

What separation properties are guaranteed in sequentially discrete spaces?

A sequentially discrete (P167) space has the property that all converging sequences are eventually constant. As a result, all such spaces are US (P99), that is, they have Unique Sequential limits. ...
Steven Clontz's user avatar
1 vote
0 answers
46 views

All separation axioms are preserved under disjoint union?

Given topological space $\{ (X_{i},O_{i}) \}_{i\in I}$, we can form their disjoint union $X := \amalg_{i}X_{i}$. Now if each $(X_{i},O_{i})$ satisfy the same separation axiom ($T_{0}, T_{1}, T_{2}, T_{...
wsz_fantasy's user avatar
  • 1,732
1 vote
1 answer
68 views

Quotients of $T_3$ spaces need not be regular.

I was reading Chapter 2 of ''Algebraic Topology for Data Scientists'' by Michel S. Postol. I had before a beginner's knowledge on Algebraic Topology and therefore wanted to at least revisit the pages ...
IAG's user avatar
  • 223
2 votes
1 answer
59 views

Does a separable, $US$, sequential space have cardinality at most the continuum?

Let $X$ be a separable sequential space with unique sequential limits ($US$). Can we prove that $X$ has cardinality at most $\mathfrak c=2^{\aleph_0}$? Context. If $X$ were Fréchet-Urysohn instead of ...
M W's user avatar
  • 9,941
7 votes
0 answers
172 views

Is a locally compact weak Hausdorff space Hausdorff?

Definitions. Let $X$ be a topological space. We say $X$ is locally compact if every member of $X$ has neighborhood basis of compact sets. weak Hausdorff if for every continuous map $f\colon Z\to X$,...
M W's user avatar
  • 9,941
2 votes
1 answer
52 views

When can we upgrade $k$-space definitions under a local Hausdorff condition?

There are a number of differing definitions of a compactly generated space, also known as a "$k$-space." Following the notation from this wikipedia article, we define a topological space $X$...
M W's user avatar
  • 9,941
4 votes
2 answers
210 views

Can a Fréchet-Urysohn hemicompact Hausdorff space fail to be locally compact?

This recent question led to some discussion on hemicompactness. A topological space $X$ is said to be hemicompact if there is an increasing sequence of compacta $K_1\subseteq\dots \subseteq K_n\...
M W's user avatar
  • 9,941
3 votes
1 answer
97 views

How to limit deformation retraction to a subset in perfectly normal spaces?

I was trying to prove some theorem in topology, which I am stuck on for quite some time. I have though of some result that seems very intuitive and would be very useful, but I can not seem to justify ...
Christian's user avatar
4 votes
1 answer
178 views

How to prove this topological lemma?

I am stuck trying to prove some result from topology. It is fairly complicated to state it here, but I realized that I could prove it, if I manage to show the following Lemma. Suppose $X$ is a normal ...
Jesus's user avatar
  • 1,798
2 votes
1 answer
72 views

Countably compact $T_0$ space $S$ such that for all $X\subseteq S$ and every $p\in X'$, there exists an open set $O$ such that ... [closed]

Is there an infinite $T_0$ space $S$ such that every infinite subset has a limit point, and such that, for all $X\subseteq S$ and every limit point $p$ of $X$, there exists an open set $O$ such that $...
Tri's user avatar
  • 417
3 votes
1 answer
95 views

Is the Alexandrov extension the unique $k_2$-Hausdorff one-point compactification of a KC space?

Definitions: KC (P100): each compact subset of the space is closed. $k_2$-Hausdorff (P171): for every compact $T_2$ space and continuous map $f:K\to X$ and points $k,l\in K$ with $f(k)\not=f(l)$, ...
Steven Clontz's user avatar
7 votes
1 answer
241 views

Where does "A family of compact sets with the finite intersection property has nonempty intersection" lie between $T_1$ and $T_2$?

This question is a follow-up of sorts to a recent question by Steven Clontz about the property that (arbitrary) intersections of compact sets are compact (abbreviated "$IKK$" in that ...
M W's user avatar
  • 9,941
0 votes
0 answers
62 views

Help understanding the Separating Hyperplane theorem.

In Chapter 2, section 2.5.1, Boyd and Vandenberghe give a proof of the Separating Hyperplane theorem. They base their proof on two points, $c \in \mathcal{C}$ and $d \in \mathcal{D}$, that are the ...
Jxson99's user avatar
  • 158
1 vote
1 answer
126 views

Let $f:X \to Y$ be a quotient map, $X$ Hausdorff. If $f$ is a proper function, then $Y$ is Hausdorff.

I've proven that $Y$ is $T_1$ using $f$ proper $\Rightarrow $ $f$ closed and since X is Hausdorff, X is $T_1$, then the unitary sets are closed in $X$. $f$ is quotient so it's surjective, then for all ...
Christian Coronel's user avatar
2 votes
1 answer
74 views

Hausdorff Spaces w/ only Trivial Continuous Maps into $\mathbb{R}$

Looking at the wiki article for the Stone-Cech compactification, it states: Andrey Nikolayevich Tikhonov introduced completely regular spaces in 1930 in order to avoid the pathological situation of ...
rubikscube09's user avatar
  • 3,925
1 vote
1 answer
53 views

Closure of limit points $\text{Cl}(\{x\}')$.

I am reviewing an article on separation axioms (Last part of Theorem 3.7, https://core.ac.uk/download/pdf/82702431.pdf ). I have trouble understanding the following statement: If $\{x\}'$ is not ...
GumK's user avatar
  • 25
2 votes
0 answers
79 views

Applications of perfectly normal spaces

One definition of a perfectly normal ($T_6$) space is that any two closed sets $A$ and $B$ can be precisely separated by a function, i.e. there is a continuous function from the whole space to the ...
A.C.'s user avatar
  • 201
56 votes
1 answer
4k views

Is there anyone among us who can identify a certain SUS space?

The property US ("Unique Sequential limits") is a classic example of property implied by $T_2$ and implying $T_1$. In fact, it's the weakest assumption out of a chain of several distinct ...
Steven Clontz's user avatar
3 votes
1 answer
82 views

Comparing two notions of "needed-for-Hausdorffness"

Suppose $\mathcal{X}=(X,\tau)$ is a non-Hausdorff space. There are two ways to make precise "the sets that need to become open if we refine $\mathcal{X}$ to become Hausdorff" that I'm ...
Noah Schweber's user avatar
0 votes
1 answer
44 views

Proof that every $T_0$ Topological Vector Space is regular

I'm currently reading through Convex Analysis and Beyond by Mordukhovich and Nam where the following preposition and proof are given (note that the authors define TVSs to be $T_0$). Proposition 1.92 ...
TheLimePixel's user avatar
1 vote
2 answers
78 views

$k_3$-Hausdorff

A subset of a space is $k_1$-closed if its intersection with every compact subset is closed in the subset. A subset of a space is $k_2$-closed if its intersection with every continuous image of a ...
Steven Clontz's user avatar
1 vote
1 answer
49 views

Are retracts always closed in a compact weak Hausdorff space?

A retract is a subspace $R\subseteq X$ for which there exists a continuous map $f:X\to R$ with $f(r)=r$ for all $r\in R$. If $X$ is compact and has the property KC (all compacts are closed, a ...
Steven Clontz's user avatar
2 votes
1 answer
64 views

Is a space $T_1$ iff every irreducible component is a point?

I've been thinking a little about irreducible components while reading Atiyah and Macdonald and wanted to try and understand the idea of irreducible component in more general terms of separation ...
Isky Mathews's user avatar
  • 3,285
6 votes
2 answers
297 views

How are k-Hausdorff and weakly Hausdorff distinct?

In this pull request to the pi-Base database, we encountered this situation. A space $X$ is said to be weakly Hausdorff provided for every compact Hausdorff space $K$ and every continuous $f:K\to X$, $...
Steven Clontz's user avatar
6 votes
1 answer
193 views

Why is a Countable Basis Needed in This Proof?

My question is regarding the Theorem in Munkres that states: Every Regular Space with a Countable Basis is Normal. Before reading the proof in Munkres, I tried to prove it myself and came up with a &...
Kenneth Winters's user avatar

1
2 3 4 5
18