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Spaces $X$ with every compactification $0$-dimensional with $\beta X\setminus X$ not locally compact

Previously, in this post I've shown the following characterization of spaces with only zero-dimensional compactifications: Theorem. Let $X$ be strongly zero-dimensional and $\beta X\setminus X$ ...
Jakobian's user avatar
  • 1,201
2 votes
1 answer
103 views

LCH spaces $X$ such that if $Y$ is a perfect image of $X$, then $Y$ is zero-dimensional

I am looking for locally compact Hausdorff spaces $X$ with the following property: If $f:X\to Y$ is a perfect map onto locally compact Hausdorff space $Y$, then $Y$ is zero-dimensional. One can see ...
Jakobian's user avatar
  • 1,201
11 votes
2 answers
314 views

Spaces with every compactification $0$-dimensional which aren't locally compact

Recently I've proven the following theorem Theorem. Let $X$ be a zero-dimensional locally compact Hausdorff space. Then the following are equivalent: Every compactification of $X$ is zero-dimensional....
Jakobian's user avatar
  • 1,201
0 votes
1 answer
145 views

Can we describe open cover compactness of a space in how the space relates to other spaces?

I've seen two definitions of connectedness of categorical flavour which I present below: (Maps into two point set): A topological space $X$ is connected iff the only continous functions $f:X \to \{ 0,...
Brian's user avatar
  • 1,525
7 votes
1 answer
134 views

Universally closed implies proper for locales

It is well known that: Theorem. For a locale (resp. topological space) $X$, the following are equivalent: $X$ is compact, i.e. every open cover of $X$ has a finite subcover. For every locale (resp. ...
Zhen Lin's user avatar
  • 15.9k
1 vote
0 answers
36 views

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

Cross-posted from Math.SE: https://math.stackexchange.com/questions/4948421/. I'm interested in comparing $k_1$-spaces, spaces whose topologies are witnessed by their compact subspaces, and $k_3$-...
Steven Clontz's user avatar
2 votes
1 answer
123 views

Signed measures on algebras (fields) and their boundedness properties

I asked this question here on math.StackEchange, but it might be too technical so I re-post it here. Let $X$ be a compact Hausdorff second countable topological space. Let $\mathcal{B}$ a countable ...
Ennio's user avatar
  • 23
0 votes
0 answers
72 views

Sequential compactness via Arzela-Ascoli theorem for uniform state spaces

Let $X$ be a uniform topological space and $C([0,1],X)$ the space of continuous functions from [0,1] to $X$. Assume that for subsets of $X$ sequential compactness and compactness are equivalent. Let $(...
PDEprobabilist's user avatar
1 vote
0 answers
75 views

Trying to achieve "some sort of hemicompactness" in a Tychonoff space

Let $X$ be a Tychonoff space, i.e. Hausdorff and completely regular. Additionally, consider a map $\psi: X \to (0,\infty)$ such that $K_R := \psi^{-1}((0,R])$ is compact in $X$, for every $R>0$. ...
Gaspar's user avatar
  • 161
1 vote
0 answers
127 views

Extremally disconnected sets as building blocks for compact Hausdorff spaces

Is every compact Hausdorff space the filtered colimit of compact extremally disconnected spaces?
Peter Kropholler's user avatar
1 vote
0 answers
76 views

Uniform approximation over compacts using weighted function spaces

I'm interested in approximations over the so-called weighted function spaces. Let $(X,\tau_X)$ be some completely regular Hausdorff topological space. Additionally, consider some map $\psi: X \to (0,\...
Gaspar's user avatar
  • 161
5 votes
0 answers
94 views

When a compact subset of a TVS can be continuously projected on a closed linear subspace?

Let $V$ be a (Hausdorff) topological vector space, $W\subset V$ a closed linear subspace, $X\subset V $ a compact. (Q): When there is a continuous map $P:X\to W$ such that $P(x)=x$ for every $x\in X\...
Pietro Majer's user avatar
  • 60.5k
11 votes
1 answer
309 views

Which closed subsets $Y$ of a compact space $X$ admit a linear extensor $C(Y)\to C(X)$?

In the following $X$ is a Hausdorff compact topological space. Let $Y$ be a closed subset of $X$. The restriction operator $R_Y:C(X)\to C(Y)$ is surjective (Tietze), so it admits a continuous right ...
Pietro Majer's user avatar
  • 60.5k
0 votes
0 answers
165 views

Are all infinite-dimensional Lie groups noncompact?

Basically what the title says — if a Lie group is infinite-dimensional, is it necessarily noncompact?
Panopticon's user avatar
6 votes
0 answers
246 views

Making the analogy of finiteness and compactness precise

If one asks about the intution behind compact topological spaces, most often one will hear the mantra “Compactness of a topological space is a generalisation of the finiteness of a set.” For example,...
Jannik Pitt's user avatar
  • 1,474
5 votes
2 answers
202 views

Polish space isometric to its hyperspace

For a Polish space $(X,d)$ its hyperspace $(K(X),d_H)$ is also a Polish space. (Here $K(X)$ denotes the set of all nonempty compact subsets of $X$, and the Hausdorff metric $d_H$ is defined by $d_H(K,...
chj's user avatar
  • 157
4 votes
1 answer
245 views

Being contained in a compact set

I have a sequential, hereditarily Lindelöf topological space $\mathcal{X}$, and some subset $A \subseteq \mathcal{X}$. I am interested in the following properties: There is some compact set $B$ with $...
Arno's user avatar
  • 4,727
6 votes
1 answer
261 views

When does base-change in topological spaces preserve quotient maps?

The question when $(-) \times X$ preserves colimits in topological spaces is well-studied. Since it always preserves arbitrary coproducts (disjoint unions), one only has to show when it preserves ...
Lennart Meier's user avatar
10 votes
1 answer
448 views

Do compactly generated spaces have a more direct definition?

Is there an elementary way to define Haussdorf-compactly generated weakly Hausdorff topological spaces in a way that does not need defining topological space first? Weakly Hausdorff sequential spaces ...
saolof's user avatar
  • 1,947
5 votes
0 answers
158 views

Does "achieving more GH-distances than some compact space" imply compactness?

Previously asked and bountied at MSE: For complete metric spaces $X,Y$, write $X\trianglelefteq Y$ iff for every complete metric space $Z$ such that the Gromov-Hausdorff distance between $X$ and $Z$ ...
Noah Schweber's user avatar
11 votes
2 answers
538 views

When is a k-space locally compact?

We're looking at the possible cardinal sequences of LCS (locally compact, Hausdorff, scattered) spaces, which has led us to think about taking a quotient of a locally compact, scattered space. A k-...
Carla Simons's user avatar
2 votes
1 answer
507 views

(Dis)prove : if every function with closed graph are continuous then the target space is compact

$(X, \tau_X) $ and $(Y, \tau_Y) $ be two topological spaces. $\forall f\in Y^X$ with $\text{Gr}(f) $ is closed implies $f\in C(X, Y) $. Question : Does this implies $(Y, \tau_Y) $ is compact? ...
SoG's user avatar
  • 307
11 votes
3 answers
890 views

Structure theorems for compact sets of rationals

Everyone knows the Heine-Borel theorem characterizing compact subsets of Euclidean space. For any $n \in \mathbb N$ a set $A \subseteq \mathbb R^n$ is compact just in case it is closed and bounded (in ...
Corey Bacal Switzer's user avatar
0 votes
1 answer
43 views

Exhaustions of product subsets by smaller product subsets

Let $X$ be a compact metric space, $A,B\subset X$ be subsets and $f\colon X\times X\to \mathbb{R}$ a continuous function that is strictly positive on $A\times B$. Do there exist increasing sequences ...
Federico Vigolo's user avatar
1 vote
1 answer
323 views

Is the restriction of a projection to a compact subset a quotient map?

Let $(X, \mathcal{T}_X)$ and $(Y, \mathcal{T}_Y)$ be topological spaces, $Z = X \times Y$, $\mathcal{T}_Z$ be the product topology on $Z$, $f : Z \to X$ be defined by $f(x, y) = x$, and $C \subset Z$ ...
kaba's user avatar
  • 397
0 votes
1 answer
199 views

Is the Čech–Stone compactification of the integers always a retract of an extremally disconnected space?

Probably $\beta \mathbb N$ is not an absolute retract (is there an easy argument for this?), but I'd be interested to know what happens in the class of extremally disconnected (compact) spaces. Is it ...
Tomasz Kania's user avatar
  • 11.3k
4 votes
0 answers
247 views

Is this property of continuous maps equivalent to some more familiar condition?

Let $f : X \rightarrow Y$ be a continuous map. Suppose that, for each collection of open sets $\{ V_i \}_{i \in I}\subset X $, $$ \bigcup_{U \subset Y \text{ open}, \ f^{-1}(U) \subset \bigcup_{i \in ...
user avatar
5 votes
1 answer
805 views

Arzelà-Ascoli for $C_b(0,1)$? Or more generally, why is that continuous functions "live most naturally" on compact spaces?

I’m wondering if there is a version of Arzelà-Ascoli for continuous functions on not-necessarily compact metric/Hausdorff spaces $X$, i.e. a characterization of the compact subsets of $C_b(X)$ (under ...
D.R.'s user avatar
  • 833
7 votes
2 answers
562 views

Is the union of a compact and the relatively compact components of its complementary in a manifold compact?

I was thinking of a way to prove this and I realised that for my approach the lemma from the title would be useful, and it´s an interesting question on its own. Obviously it is true if the manifold is ...
Saúl RM's user avatar
  • 10.6k
7 votes
1 answer
899 views

Is a closed subset of an extremally disconnected set again extremally disconnected?

Let $T$ be a compact Hausdorff extremally disconnected set (so $T$ is a compact Hausdorff space, such that the closure of each open subset is again open). Let $S \subseteq T$ be a closed subset. ...
AlexIvanov's user avatar
0 votes
2 answers
546 views

A question about locally compact spaces

Recently I read a book about linear algebraic group written by Ian Macdonald. There is a conclusion which I can't prove. It says that if $X$ is locally compact Hausdorff space, then $X$ is compact if ...
Fuutorider's user avatar
5 votes
1 answer
298 views

Can Tychonoffs theorem for a countable number of spaces be proven with ZF plus the axiom of (countable) dependent choice?

It can be proven without any form of infinite choice that the product of two compact spaces (and thus any finite product) is compact, while on the other hand, it is well known that the general form of ...
saolof's user avatar
  • 1,947
5 votes
1 answer
168 views

Compactness of symmetric power of a compact space

Suppose I have a compact metric space $(X,d)$ and let $\mathcal{X}=X^K$ be the product space. Consider the equivalence relation $\sim$ on $\mathcal{X}$ given as: for $\alpha,\beta\in \mathcal{X}$, $\...
Sunrit's user avatar
  • 59
1 vote
0 answers
540 views

Is the set of compact operators closed with the strong topology?

It is well-known that the space of compact operators over Banach spaces is closed within the norm topology. My question: Let $X$ be a Banach space. Considering the strong topology (defined by ...
Malik Amine's user avatar
1 vote
1 answer
137 views

Density and compactness of Boolean embeddings

Let A and B be Boolean algebras and $h:A\rightarrow B$ a Boolean embedding. If every element of $B$ can be expressed both as a join of meets and as a meet of joins of elements in $h(A)$, then the ...
IJM98's user avatar
  • 281
1 vote
0 answers
137 views

Relative compactness... but what is the toplogy?

The following Theorem was described in a text I was reading as a compactness result. The proof is probably too advanced for me but I was just wondering with respect to what topology we have ...
edamondo's user avatar
  • 111
8 votes
1 answer
272 views

Characterization of pretty compact spaces

This is a cross post from MSE. I believe that the following problem have already been considered by some sophisticated topologist. Definition 1. A non-compact Hausdorff topological space $X$ is called ...
Norbert's user avatar
  • 1,697
3 votes
2 answers
485 views

Relative compactness in topological spaces (reference request)

Motivation and context: For a subset $S$ of a metric space $(M,d)$, the following are two very classical compactness results in Analysis: 1a) The set $S$ is compact if and only if each sequence in $S$...
Jochen Glueck's user avatar
4 votes
2 answers
283 views

Compact spaces whose compactness does not come from a product of compact spaces

For the (Hausdorff) compact spaces I can think of, compactness is established either using a product of compact spaces (including the Heine-Borel Theorem, the Banach-Alaoglu Theorem, Stone-Čech ...
Cameron Zwarich's user avatar
2 votes
0 answers
2k views

On weak compactness of the unit ball in a reflexive Banach space

It is a well known result in functional analysis that a Banach space $X$ is reflexive if and only if the unit ball is weakly compact (compact in the weak topology). This result is also known as ...
Arian's user avatar
  • 364
0 votes
1 answer
148 views

About the finished, $\aleph_0$...-compactness

Definitions : $(E,d)$ a metric space is finished-compact if any covering of $E$ by open, we can extract a finite subcover $(E,d)$ is $\aleph_0$-compact if for any infinite covering of $E$ by open, we ...
Dattier's user avatar
  • 4,074
1 vote
0 answers
166 views

Subspaces of compact spaces and quotients of Hausdorff spaces

Let $\operatorname{Top}$ be the class of topological spaces. Furthermore, let $\mathcal{U}\subset\operatorname{Top}$ and $\mathcal{V}\subset\operatorname{Top}$ classes satisfying the following ...
cl4y70n____'s user avatar
6 votes
1 answer
582 views

When Stone–Čech compactification is totally disconnected

A topological space $X$ is totally disconnected if the connected components in $X$ are the one-point sets, and a topological space, $X$ is called completely regular exactly in case points can be ...
Arena's user avatar
  • 61
6 votes
1 answer
641 views

Uniqueness of limits and compactness implies closure

It is not difficult to prove that in a Hausdorff topological space every compact set is closed, and almost trivial that if in a topological space X every compact set is closed then X is T1. As ...
Daniel Elessar's user avatar
11 votes
0 answers
273 views

A ZFC-example of a countably compact paratopological group which is not a topological group

Problem. Does there exist a ZFC-example of a countably compact Hausdorff paratopological group which is not a topological group? (The problem posed 27 May 2015 by Alexander Ravsky on page 9 of Volume ...
Lviv Scottish Book's user avatar
1 vote
2 answers
529 views

Extending homeomorphisms between compact metric subsets

Let $X$ be a compact metric, second countable space with finite covering dimension. Let $A,B$ be two closed subsets of $X$. Assume that $h:A\to B$ is a homeomorphism. Is it possible to extend $h$ to a ...
Betti's user avatar
  • 11
6 votes
1 answer
307 views

Is there a compactification with nontrivial connected remainder?

Question: Let $X$ be a continuum and $p \in X$. Under what conditions does there exist a compactification $\gamma (X-p)$ with $\gamma (X-p) - (X-p)$ connected and nondegenerate? Throughout, $X$ is a ...
Daron's user avatar
  • 1,955
3 votes
1 answer
331 views

Checking finite subcover property on dense subset

Let $X$ be a topological space with a dense subset $D\subseteq X$. Suppose that every open cover of $X$ has a finite subfamily which covers $D$. Can I conclude that $X$ itself is compact? The answer ...
Tobias Fritz's user avatar
  • 6,406
12 votes
1 answer
2k views

What are compact objects in the category of topological spaces?

Let $\mathscr C$ be a locally small category that has filtered colimits. Then an object $X$ in $\mathscr C$ is compact if $\operatorname{Hom}(X,-)$ commutes with filtered colimits. On the other hand, ...
R. van Dobben de Bruyn's user avatar
5 votes
2 answers
852 views

Covering compactness in the weak sequential topology

Let $X$ be a real Banach space. Apart from the norm topology, we can consider the following weak topologies on $X$: the weak toplogy, defined as the initial topology with respect to $X^*$. In other ...
Daniel Steck's user avatar