All Questions
Tagged with compactness gn.general-topology
80 questions
1
vote
2
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202
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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 $\beta X\setminus X$ ...
- 1,201
2
votes
1
answer
103
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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 ...
- 1,201
11
votes
2
answers
314
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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....
- 1,201
0
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1
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145
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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,...
- 1,525
7
votes
1
answer
134
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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. ...
- 15.9k
1
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0
answers
36
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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 $k_3$-...
- 1,322
2
votes
1
answer
123
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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 ...
- 23
0
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0
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72
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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 $(...
- 133
1
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0
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75
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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 $R>0$. ...
- 161
1
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0
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127
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Extremally disconnected sets as building blocks for compact Hausdorff spaces
Is every compact Hausdorff space the filtered colimit of compact extremally disconnected spaces?
- 1,638
1
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0
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76
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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,\...
- 161
5
votes
0
answers
94
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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\...
- 60.5k
11
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1
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309
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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 ...
- 60.5k
0
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0
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165
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Are all infinite-dimensional Lie groups noncompact?
Basically what the title says — if a Lie group is infinite-dimensional, is it necessarily noncompact?
- 194
6
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0
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246
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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,...
- 1,474
5
votes
2
answers
202
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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,...
- 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 $...
- 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 ...
- 12.1k
10
votes
1
answer
448
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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 ...
- 1,947
5
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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 $Z$ ...
- 21.2k
11
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2
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538
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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-...
- 111
2
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1
answer
507
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(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?
...
- 307
11
votes
3
answers
890
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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 ...
- 1,882
0
votes
1
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43
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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 ...
- 380
1
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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 $C \subset Z$ ...
- 397
0
votes
1
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199
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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 ...
- 11.3k
4
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0
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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 ...
user30211
5
votes
1
answer
805
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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 ...
- 831
7
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2
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562
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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 ...
- 10.6k
7
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1
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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.
...
- 755
0
votes
2
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546
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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 ...
- 89
5
votes
1
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298
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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 ...
- 1,947
5
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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 $\alpha,\beta\in \mathcal{X}$, $\...
- 59
1
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0
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539
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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 ...
- 299
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 ...
- 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 ...
- 111
8
votes
1
answer
272
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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 ...
- 1,697
3
votes
2
answers
485
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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 $S$...
- 12.6k
4
votes
2
answers
282
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 ...
- 3,816
2
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0
answers
2k
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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 ...
- 364
0
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1
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148
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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 ...
- 4,074
1
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0
answers
166
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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 ...
- 353
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 ...
- 61
6
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1
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641
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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 ...
11
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0
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273
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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 ...
- 4,535
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 ...
- 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 ...
- 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 ...
- 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, ...
- 28.7k
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 ...
- 171