All Questions
Tagged with gn.general-topology compactness
65
questions
0
votes
0
answers
152
views
Are all infinite-dimensional Lie groups noncompact?
Basically what the title says — if a Lie group is infinite-dimensional, is it necessarily noncompact?
- 183
6
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0
answers
186
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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,...
- 1,093
5
votes
2
answers
165
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,...
- 157
4
votes
1
answer
208
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 $...
- 4,096
6
votes
1
answer
180
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 ...
- 11.9k
9
votes
1
answer
396
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 ...
- 1,803
0
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0
answers
46
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Perfectly normal compactification of weak-star dual of Banach space
Let $X$ be an infinite-dimensional (otherwise the answer to my question below is trivial) separable real Banach space with topological dual $X^*$, and denote by $\sigma(X^*,X)$ the weak-star topology ...
- 101
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0
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149
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 $Z$ ...
- 19k
11
votes
2
answers
402
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-...
- 111
2
votes
1
answer
448
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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?
...
- 317
9
votes
3
answers
691
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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 ...
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1
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42
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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 ...
- 390
1
vote
1
answer
250
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 $C \subset Z$ ...
- 387
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1
answer
152
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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 ...
- 11.1k
5
votes
0
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235
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 ...
- 4,647
4
votes
2
answers
545
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 ...
- 569
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2
answers
440
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 ...
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6
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1
answer
811
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.
...
- 735
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votes
2
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378
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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 ...
- 99
4
votes
1
answer
223
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 ...
- 1,803
5
votes
1
answer
154
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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
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0
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422
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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 ...
- 289
1
vote
1
answer
129
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 ...
- 261
1
vote
0
answers
136
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
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votes
1
answer
251
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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 ...
- 1,627
3
votes
2
answers
302
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 $S$...
- 11.5k
4
votes
2
answers
269
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 ...
- 3,222
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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
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votes
1
answer
143
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 ...
- 3,609
1
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0
answers
153
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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 ...
- 343
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1
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522
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 ...
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1
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545
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 ...
11
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252
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 ...
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vote
2
answers
446
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
284
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 ...
- 1,761
3
votes
1
answer
301
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 ...
- 5,596
11
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, ...
- 26.5k
5
votes
2
answers
789
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
9
votes
2
answers
761
views
On the definition of locally compact for non-Hausdorff spaces
It seems that there are different conventions in the literature as to what is a locally compact space (when the space is not supposed Hausdorff).
The two main non equivalent definitions I've seen ...
- 975
13
votes
1
answer
1k
views
A topology on $\Bbb R$ where the compact sets are precisely the countable sets
QUESTION.
In there a topology on $\Bbb R$ where the compact subsets are precisely the countable subsets?
I am trying to create a counterexample to a certain claim, and I found that what I need is a ...
- 233
8
votes
3
answers
580
views
Is there a non-metrizable topological space for which any countably compact subset is compact?
The title is the question : Is there a non-metrizable topological space for which any countably compact subset is compact ?
EDIT : non-metrizable and Hausdorff
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6
votes
2
answers
2k
views
How do I prove that compact-open topology is metrizable?
Let $X$ be a $\sigma$-compact topological space and $(Y,d)$ be a metric space.
Let $\{K_n\}$ be a sequence of compact subsets of $X$ whose union is $X$.
Define $\rho_n(f,g):=\sup \{d(f(z),g(z)): z\...
- 337
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votes
2
answers
2k
views
Is there an easier proof to show that the closed convex hull of a normalized weakly null sequence is weakly compact?
In a paper that I am reading there is a following step:
Let $X$ be a Banach space and let $(x_k) \subset X$ be a normalized sequence that converges weakly to $0$.
Then $\overline{co}(x_k)$ is a ...
- 163
1
vote
0
answers
124
views
Category-theoretic characterization of zero-dimensional spaces
Some background: a zero-dimensional space is one admitting a basis of clopen sets, whereas an extremely disconnected space is one where the closures of open sets are open. In the category CHauss of ...
- 369
2
votes
1
answer
206
views
A quasicompact space with a net that contains no convergent strict subnet
If $x:\Lambda \rightarrow X$ is a net in a topological space $X$ and $\Lambda '\subseteq \Lambda$ is a cofinal subset of the directed set $\Lambda$, then $x|_{\Lambda '}$ is a subnet of $x$. We call ...
- 1,220
2
votes
0
answers
94
views
On compactness in $C(X)$
Let $X$ be a Tychonoff space. It is well known, that for a family of scalar functions equicontinuity + pointwise boundedness imply relative compactness in $C(X)$ (with compact-open topology). It is ...
- 5,275
3
votes
0
answers
183
views
Which compact topological spaces are homeomorphic to their ultrapower?
It is well known that for any compact metric space $(X, d)$, and any ultrafilter $\mu$ there is a map $i_\mu:\prod_\mu (X, d) \to (X_d)$ in the category of metric spaces and Lipschitz maps where $i_\...
- 2,209
3
votes
1
answer
794
views
On compactness in Sion's minimax theorem
Sions minimax theorem (wiki, paper) can be stated as follows:
Let $X$ be a compact convex subset of a linear topological space and $Y$ a convex
subset of a linear topological space. Let $f$ be a ...
- 349
2
votes
0
answers
182
views
Regularity of Dirac measure on Baire sets [closed]
Suppose $X$ is a locally compact Hausdorff space.
Define the Baire sets in $X$, denoted by $\mathcal Ba(X)$,
to be the smallest $\sigma$-algebra that contains all compact $G_\delta$ subsets of $X$.
...
- 363
6
votes
2
answers
567
views
Anti-compactness
Let $(X,\tau)$ be a topological space such that
$\tau\ne\{\emptyset\ X\}.\ $
We call an open cover $\mathcal{U}$ of $(X,\tau)$ proper if
$\ X\notin \mathcal{U}.\ $ Moreover we say that $(X,\...
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