Questions tagged [compactness]

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6
votes
1answer
118 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 ...
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 ...
2
votes
1answer
56 views

Compact embedding of space of signed Radon measures into Sobolev space $W^{-1,q}$ from Evans paper; Does it work in one space dimension?

Background: I work on a PDE problem where I have some approximating sequence of measure-valued functions and I need to compactly embed it into some negative Sobolev space $W^{-m,q}$ on the bounded ...
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 ...
0
votes
0answers
116 views

Spectral Theorem for compact self-adjoint operators on real Hilbert spaces [duplicate]

Is the spectral theorem for self-adjoint compact operators on a Hilbert space also true if the Hilbert space is real (instead of complex)? Wikipedia says this is true. However, it seems to me that ...
6
votes
2answers
486 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 $K$....
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 ...
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 ...
0
votes
1answer
217 views

Can I get away without using Arzela-Ascoli?

I am currently thinking of function-valued random variables. In order to prove a result, I need to approximate by (function-valued) step functions. This naturally leads to the idea of chopping up the ...
0
votes
0answers
247 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 ...
0
votes
1answer
110 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 ...
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 ...
1
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0answers
96 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 ...
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 ...
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(\...
0
votes
1answer
64 views

Finite sub-covering of ball in Hilbert space

Let $H$ be a separable Hilbert space, $B$ the closed unit ball in $H$, and $(x_n)$ be a dense sequence in $B$, $(e_n)$ be an orthonormal basis, and $\varepsilon>0$. Then $B$ is covered by the (...
-1
votes
1answer
88 views

Compactness of a special kind of Integral operators

Let $(S(t))_{t>0}$ be a continuous operator from $L^2(0,1)$ to its self and Let $K$ be the operator $$\eqalign{ & K:{L^2}(0,1) \to {L^2}(0,1) \cr & f: \to (Kf)(x) = \int\limits_0^1 {k(...
2
votes
0answers
82 views

Approximation of deterministic problems in the PDEs with the stochastic ones

A lot of problems in PDE theory are solved in the following way: The original problem is quite hard and we can't solve it, so we make the approximation problem that we can solve. Than we go back and ...
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
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 ...
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 ...
0
votes
0answers
115 views

Compact operator

Let $k:[0,1]^2 \to [0,1]$ be a measurable function. Define $K:L^2([0,1])\to L^2([0,1])$ to be the operator: $$ (Kf)(x) = \int_0^1\int_0^1 f(z) k(x,y) \mathbf{1}_{x\leq z\leq y} \ \mathrm{d}z \mathrm{d}...
1
vote
0answers
48 views

Continuous injection of metric ball into Euclidean ball

This is a follow-up to this post. Suppose that $(X,d_X)$ is a compact metric space with (finite) metric-capacity, defined by $$ \kappa_X(\epsilon)\triangleq\sup\left\{ k : \exists x_0,\dots,x_k \...
5
votes
1answer
178 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 ...
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 ...
5
votes
1answer
306 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 ...
2
votes
1answer
139 views

“Compactness in Measure” in Function Spaces

In Chapter 4.9 of the book "Measures of Noncompactness and Condensing Operators" (Vol. 55 of Operator Theory: Advances and Applications), the authors mention the property "compactness in measure". ...
4
votes
2answers
185 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 ...
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 ...
1
vote
2answers
176 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 ...
1
vote
0answers
152 views

Compact embedding result

Let $\tau$ and $\ell$ be positive numbers. We know that the space $H^2(0,\ell)\cap H^1_0(0,\ell)$ is compactly embedded into $L^6(0,\ell)$. Now, is the space $L^2(0,\tau;H^2(0,\ell)\cap H^1_0(0,\ell))$...
0
votes
2answers
520 views

Does point-wise weak convergence give weak convergence in $L^2(I;X)$?

Let $X$ be a separable reflexive Banach space, $F$ be a locally Lipschitz nonlinear operator on $X$ that is weakly continuous on $X$, and $u_n$ are $u_n$ weakly converges to $u$ on $L^2(0,T;X)$. Now, ...
0
votes
1answer
47 views

Does $K^{1/2} (t,s)$ inherit the continuity of $K(t,s)$?

Assume that $K(t,s)$ is a (1) symmetric, (2) continuous, and (3) positive definite kernel on $[0,1] \times [0,1]$. The spectral decomposition of $K(t,s)$ is: $$ K (t,s) = \sum_{i=1}^\infty \lambda_i \...
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 ...
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 ...
1
vote
1answer
97 views

Extend a bundle “trivially”

Suppose I have a fibre bundle $E\to B$ with compact fibre. Furthermore, $B$ is open in a larger, compact space, e. g. $B\subseteq B'$. I want to get a map $E'\to B'$ (not a bundle any more!) with $E'|...
2
votes
1answer
134 views

Li-Yorke chaos: the non compact case

1) Is there any notion of Li-Yorke chaos for non compact (metric) spaces $X$ and non continuous transformation $f:X \rightarrow X$? Could you bring some references? 2) I mean, why are so important ...
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 ...
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 $f,h \in L^1$ ...
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 ...
0
votes
1answer
291 views

Upper bound for KL divergence on compact space

Let $(\Omega, \mathcal{F}, \mu)$ be a probability space and let $Q$ be the uniform distribution on $(\Omega, \mathcal{F})$ such that $q = dQ / d\mu$ exists. Then the KL-divergence for some probability ...
1
vote
1answer
252 views

Riemannian manifolds: every compact subset is contained in a connected relatively compact open subset [closed]

While working on some problem (not relevant here), it turned out to be convenient to be able to enclose arbitrary compact subsets in "nicer" compact subsets, hence the question: if $(M,g)$ is a ...
-6
votes
1answer
222 views

How do i show that every compact complex manifolds of dimension one admit nonconstant meromorphic function? [closed]

let $X$ be a compact complex manifold of dimension one . Question: How do i show that every compact complex manifolds of dimension one admit nonconstant meromorphic function ? . ...
1
vote
3answers
510 views

How to show the cardinality of nonisometric compact metric spaces is the continuum

It is asserted in A Course in Metric Geometry by Burago, Burago, Ivanov that there can be no more than continuum of mutually nonisometric compact spaces How is this proven? Its clear that there ...
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 ...
8
votes
1answer
871 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, ...
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, ...
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 ...
3
votes
1answer
355 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 ...
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,\...