Questions tagged [compactness]

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27
votes
1answer
795 views

Can closed compacts in a topological group behave “paradoxically” with respect to unions, intersections, and one-sided translations?

Consider two closed compacts $A$ and $B$ in a topological group $\Gamma$. Let $A'$ be a left translation of $A$ and $B'$ a left translation of $B$: $A' = aA$, $B' = bB$. Suppose it is known that $A'\...
15
votes
2answers
731 views

Does there exist a supercompactness theorem?

Large cardinals such as weakly compact cardinals, measurable cardinals, strongly compact cardinals, and extendible cardinals all can be characterized in terms of a certain compactness theorem of ...
13
votes
1answer
611 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 ...
13
votes
1answer
471 views

A generalization of the Arhangelskii Theorem

Arhangeleskii's Theorem states the following For any Hausdorff topological space $X$, $$ |X|\leq2^{\chi(X)L(X)} $$ where $\chi(X)$ is the character of $X$ and $L(X)$ is the Lindelöf degree of $...
11
votes
0answers
193 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 ...
9
votes
0answers
148 views

Masas in SAW*-algebras

I asked this question three years ago at MSe but it has no response; let me try here. Pedersen distilled the following class of C*-algebras which he termed SAW*-algebras (Journal of Operator Theory, ...
8
votes
3answers
481 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
8
votes
2answers
541 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 ...
8
votes
1answer
872 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, ...
8
votes
2answers
346 views

Compact operators on Lebesgue spaces

Let $K:{\rm L}^p({\bf R}^d)\to {\rm L}^p({\bf R}^d)$ be a bounded linear operator for every $p\in(1,\infty)$. Assume that for some $r\in(2, \infty)$ it holds that $K$ is compact on ${\rm L}^q({\bf R}^...
8
votes
2answers
472 views

Totally disconnected subspaces

This question is motivated by this one, where no simple solution within ZFC seems to exist. Let me ask a weaker question then. Suppose that $K$ is a compact, Hausdorff, non-metrizable space. Does it ...
7
votes
1answer
262 views

Compactness of set of indicator functions

Let $\chi_A(x)$ denote an indicator function on $A\subset [0,1]$. Consider the set $$K=\{\chi_A(x): \text{ A is Lebesgue measurable in }[0,1]\}.$$ Is this set compact in $L^\infty(0,1)$ with respect ...
6
votes
14answers
4k views

Applications of compactness [closed]

Similar to this question: Applications of connectedness I want to collect applications of compactness. E.g.: compact + discrete => finite, which can be used to prove the finiteness of the ...
6
votes
2answers
497 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,\...
6
votes
2answers
252 views

Is the separability of the space needed in the proof of the Prohorov's theorem?

The Section 5 of the book: Billingsley, P., Convergence of Probability Measures, 1999, studies Prohorov's theorem. A short reminder is given below. Let $\Pi$ be a family of probability measures on ...
6
votes
1answer
119 views

Does a compact Lie group have finitely many conjugacy classes of maximal Abelian Lie subgroups?

Let $G$ be a compact Lie group. An Abelian Lie subgroup $A \leq G$ is a maximal Abelian Lie subgroup if, for any Abelian Lie subgroup $A'$ such that $A \leq A' \leq G$, then $A' = A$. Of course any ...
6
votes
2answers
487 views

CG spaces from the perspective of sheaves over compact Hausdorff spaces

A compactly generated space is a space $X$ such that $f : X \rightarrow Y$ is continuous if and only if $K \rightarrow X \stackrel{f}{\rightarrow} Y$ is continuous for each compact hausdorff space $K$....
6
votes
1answer
409 views

Weak*-convergence of signed measures

Let $X$ be a compact Hausdorff space and let $M(X)$ denote the space of signed measures that is naturally dual to $C(X)$, the space of continuous functions on $X$. I am interested whether the ...
6
votes
1answer
230 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 ...
5
votes
2answers
494 views

Beyond Cantor's Teepee

From Counterexamples in Topology by Steen and Seebach (2nd edition) example 129 page 145 we have an example of connected and totally path-disconnected space. It is defined as follow: Fix $p= (1/2,1/2)...
5
votes
1answer
232 views

Existence of injective compact operators

We know that if $X$ is a separable Banach space, then for every infinite dimensional Banach space $Y$, there exists an injective compact operator from $X$ to $Y$. My query is for every Banach ...
5
votes
1answer
309 views

Mapping scheme from a proper variety

Let $X$ be a proper scheme over a field $k$. Let $T$ be a scheme over $k$. Is it true that morphisms $T \times X \to \mathbb{A}^1$ are in bijection with morphisms $T \to \Gamma (X, \mathcal{O}_X)$ (...
5
votes
1answer
583 views

Rellich-Kondrachov compacteness Theorem for the Euclidean space with Gaussian measure

Let $\gamma_n: \mathbb{R}^n\to\mathbb{R}$ be the Gaussian distribution function defined by $$ \gamma_n(x):=(2 \pi)^{-\frac{n}{2}} e^{-\frac{|x|^2}{2}}. $$ Let $d\gamma_n$ denote the following measure ...
5
votes
2answers
460 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 ...
5
votes
1answer
176 views

Corson-Lindenstrauss : Weakly compact sets as intersection of finite unions of cells

A theorem of Corson and Lindenstrauss in: Corson, H. H. and Lindenstrauss, J. “On weakly compact subsets of Banach spaces”. In: Proceedings of the American Mathematical Society 17.2 (1966), pp. 407–...
5
votes
1answer
403 views

Locally finite compact groups

I assume all tolpological groups here to be Hausdorff. A group is called locally finite if every finitely generated subgroup is finite. What can be said about a locally finite compact group? Must it ...
5
votes
1answer
307 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 ...
5
votes
1answer
179 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 ...
5
votes
0answers
301 views

Compact quasi-coherent sheaves

Let $X$ be a scheme. What are the compact objects in the category of quasi-coherent $\mathcal{O}_X$-modules? All references seem to discuss the derived category but I need the abelian category.
5
votes
0answers
225 views

Pullback of Morse form satisfies Palais Smale

Let $(\alpha,g)$ be a Morse-Smale pair on a closed smooth manifold $M$, i.e. $\alpha$ is a Morse form and $g$ a Riemannian metric on $M$ such that stable and unstable manifolds of the gradient vector ...
5
votes
0answers
405 views

Noncompactness of the Sobolev embedding in the critical exponent case

Let $\Omega \subset \mathbb R^n$ be a bounded domain with a Lipschitz boundary and $n > p \ge 1$. It is well known that up to the critical exponent $p^* = pn/(n − p)$, i.e. $q < p^*$, the ...
5
votes
0answers
248 views

trace-class embeddings

There is a classical theorem of Riesz-Kolmogorov that characterizes compact embedding in $L^p$-spaces of some subspace of them. A generalization to arbitrary metric spaces has been recently obtained ...
4
votes
3answers
374 views

Can the intersection of the boundaries of compact and convex sets be a single element?

Let $H_1,H_2,\dots,H_n$ be compact and convex sets in $\mathbb{R}^n$ such that $\bigcap_{j=1}^n H_j$ has non-empty interior and for each $i=1,2,\dots,n$ there exist at least one element $x \in H_i$ ...
4
votes
2answers
806 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 ...
4
votes
2answers
331 views

When is a Nemytskii map between Sobolev spaces compact?

Let $f:\mathbb{R} \to \mathbb{R}$ be a smooth function with bounded derivative. Define the Nemytskii map $F:H^1(\Omega) \to H^1(\Omega)$ by $F(u)(x) := f(u(x))$. Here $\Omega$ is a bounded smooth ...
4
votes
2answers
186 views

Compact images of nowhere dense closed convex sets in a Hilbert space

Let $B=-B$ be a nowhere dense bounded closed convex set in the Hilbert space $\ell_2$ such that the linear hull of $B$ is dense in $\ell_2$. Question. Is there a non-compact linear bounded operator ...
4
votes
2answers
1k 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\...
4
votes
1answer
483 views

Fréchet-Kolmogorov compactness Theorem for Lp spaces on manifolds

Suppose I have a family of functions $\mathcal{F} \subseteq L^2(\mathcal{M}, P)$ where $\mathcal{M}$ is a compact manifold, and $P$ is a probability distribution on $\mathcal{M}$. Is there an ...
4
votes
2answers
195 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 ...
4
votes
1answer
252 views

Weak compactness of order intervals in $L^1$

Let $(\Omega,\mu)$ be a measure space, say $\sigma$-finite for the sake of simplicity, and let $L^1 := L^1(\Omega,\mu)$ denote the real-valued $L^1$-space over $(\Omega,\mu)$. For all $f,h \in L^1$ ...
4
votes
1answer
126 views

Reeb stability counterexample: foliation in $S^{n-2}\times S^1\times S^1$ with non-diffeomorphic leaves

Reeb's global stability theorem requires the foliation to be of codimension 1. As a counterexample, in "Geometric theory of foliations", Camacho and Lins Neto present the following. Consider the ...
3
votes
2answers
302 views

Are compact sets in a Banach lattice order bounded?

Given a compact subset $A$ of a Banach lattice $E$, is the following true? There exist $u,v\in E$ so that $u\leq a\leq v$ for all $a\in A$. This is true in case $E=C(X)$, $X$ compact, with the ...
3
votes
3answers
728 views

Compactness of sigma-algebra for the $L^1$ metrics

Consider a probability space $(X,F,\mu)$, and the quotient $G$ of the sigma-algebra $F$ by its null sets. Endow $G$ with the metric $d(A,B) = \mu(A \triangle B)$. Is $(G,d)$ a compact metric space? ...
3
votes
1answer
125 views

Bounded ball measure on compact metric space

Fix $c>1$. Let $(X,d)$ be a separable compact metric space, does there necessarily exist a Borel probability measure $\nu$ on $(X,d)$ such that $\operatorname{sup}_{x \in X,r>0}\frac{\nu(\...
3
votes
1answer
166 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 ...
3
votes
1answer
356 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 ...
3
votes
1answer
116 views

A non-condensing operator with a power condensing

Let $\alpha$ to be the Kuratowski measure of non-compactness, in a Banach space $E$. It's very easy to prove that $\alpha (D_1\times D_2)\leq \alpha (D_1)+\alpha (D_2)$, where $D_1$ and $D_2$ are ...
3
votes
0answers
101 views

Approximation argument in geometric flows

I'm studying by myself Mean Curvature Flow and I'm reading the paper "Interior estimates for hypersurfaces moving by mean curvature" by Klaus Ecker and Gerhard Huisken and I'm stuck in the following ...
3
votes
0answers
83 views

Compact generation of the category of D-modules on moduli stack of principal bundles for algebraic groups?

Let $k$ be an algebraically closed field of characteristic 0. Let $X$ be a connected smooth complete curve over $k$. Consider the moduli stack $\mathrm{Bun}_G$ of principal $G$-bundles on $X$ for ...
3
votes
0answers
220 views

Compact embedding of ${\rm L}^1_{loc}$ space

I was reading one preprint and stumbled upon a part in the proof where one particular embedding was used. Namely: Let $\Omega\subset{\bf R}^2$ be open and bounded and let $p\in\langle 1,2\rangle$. ...