Questions tagged [compactness]
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203 questions
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Logic-Compactness in complete lattice
Assume $\Gamma$ is an infinite set composed of formulas (of finite length), and $A$ is a formula (of finite length). For example, $\Gamma=\{x,y\sqcup z,x_1,x_2,x_3,\cdots\}$, $A=(x\sqcap y)\sqcup(x\...
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Sections with compact support commutes with tensor over a locally compact Hausdorff space. Why can we reduce to the compact case?
(I originally asked this in MSE but since there I didn't receive any feedback I repost it to MO.)
I have a question about the proof of the following result in Kashiwara, Schapira, Sheaves on Manifolds:...
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A relation between maximal subgroups of a left topological semigroup
Suppose $S$ is a left simple compact $T_2$ left topological semigroup (meaning $s\mapsto ss_0$ is continuous for each $s_0\in S$ -- in this setting, left simplicity implies that it is actually a ...
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Is the projection of a Borel set Haar-measurable?
Suppose $G_1,G_2$ are compact $T_2$ groups (if it helps, we can assume that $G_1=G_2^n$ for some $n\in\mathbf N$) and suppose $B\subseteq G_1\times G_2$ is Borel (if it helps, we can assume that $B$ ...
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Composition and compactly generated spaces
Let $X$ and $Y$ and $Z$ be compactly generated Hausdorff spaces. Is it true that the composition map $Z^Y \times Y^X \rightarrow Z^X$ is continuous? Here the function spaces are given the compact-...
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Weak compactness criterion in $L^\infty (0,1; L^1 (\mathbb{R}^n))$?
I was wondering wether there are simple weak compactness criterions for sequences in $L^\infty (0,1; L^1 (\mathbb{R}^n))$. The space $L^1$ is not a dual so weak-$*$ compactness is ruled out, and there ...
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Refinement of right order topology on analytic subsets of $\mathbb R$
Follow on from this question and this question, with the aim of solving this question.
Let $\mathcal C\subseteq \mathbb R$ denote the Cantor set and let $A\subseteq \mathcal C$ be dense, and analytic. ...
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Locally compact topology on subsets of Cantor set
Cross-posting this question from MSE.
Let $C\subseteq \mathbb R$ denote the Cantor set and let $A\subseteq C$ be dense with $|A| = |\mathbb R|$, and assume $A$ doesn't contain its infimum. Is there a ...
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Locally compact, second countable, Hausdorff topology refining the right order topology
Let $A\subseteq \mathbb R$ be Borel. Is there a second countable, locally compact, Hausdorff topology on $A$ that is finer than the right order topology (i.e. the topology with base $\{x\in A:a< x\}...
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Disjoint bends touching circles
Let $X$ be a compact subset of $\mathbb R^2$ and $x\in X$.
It is known that if every circle centered at $x$ contains an arc within $X$, then $X$ contains some region (filled disc) of the plane.
Proof. ...
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Ergodic automorphisms of compact groups are isomorphic to Bernoulli shifts
In the paper [A] the author proves that ergodic automorphisms of compact groups are isomorphic to Bernoulli shifts. However the proof is discussed only in the abelian case and the last phrase of the ...
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Hausdorff compact spaces that admit a Borel measure with full support?
All separable spaces admit Borel probability measures with full support (e.g. a series of delta’s). The one-point compactification of an uncountable space does not have such a measure (the Borel $\...
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Gluing compact $F$-spaces along a $P$-set is an $F$-space
The lemma 1.4.1 in van Mill's Introduction to $\beta \omega$ says that if $X, Y$ are compact $F$-spaces and $f:A\to Y$ is continuous where $A\subseteq X$ is a $P$-set, then $X\cup_f Y$ is an $F$-space ...
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Proof of Theorem 15 in Protasov's work on inductively compact groups
I am currently studying the properties of inductively compact groups, and I came across Theorem 15 in Protasov's work, which states that for every locally compact group
G, the set of inductively ...
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Sizes of compact, separable spaces / compactifications of the integers - part 2
This is a follow-up of my previous question.
Again, all topological spaces are Hausdorff.
Let $\kappa$ be a cardinal.
From there recall:
Define $CS(\kappa) \Leftrightarrow$ there exists a compact, ...
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Sizes of compact, separable spaces / compactifications of the integers
This is a cross-posting from my MSE question.
See also part 2 here.
In the following, all topological spaces are Hausdorff.
Let $\kappa$ be a cardinal.
Define $CS(\kappa) \Leftrightarrow$ there ...
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1
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Compactness in trace class operators space
Let $H$ be a separable Hilbert space. Let $L_1$ denote the space of trace class operators on $H$ with the trace-class norm $\|\cdot\|_1$, i.e. $\|K\|_1=Tr|K|$ for all $K\in L_1$.
Are there easy ...
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Question about coverings of zero Hausdorff measure compact sets
Consider a compact set $E\subset \mathbb{R}^n$ ad assume that ${\cal H}^{n-1}(E)=0$.
If $\epsilon>0$, there is a finite covering of $E$ made of $N_\epsilon$ open balls $B_{r_{\epsilon,k}}(x_{\...
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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 $\beta X\setminus X$ ...
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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 ...
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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....
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Compactness with respect to topology induced by total-variation distance
I've been working on a problem and at some point in the proof I need to show that the following set $$\left\{\mu \in \mathcal{P}_{ac}(\mathbb R^d): \int \varphi(x)\mu(\mathrm{d}x)\leq C\right\}$$
is ...
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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,...
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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. ...
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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 $k_3$-...
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Proving that compact Hausdorff groups are cofiltered limits of compact Lie groups
What is the easiest way to show that a compact hausdorff topological group is a closed subgroup of a product of finite dimensional Lie groups?
Here are the relevant definitions:
Definition: (compact ...
user531936
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0
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318
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Embeddings of Bochner-Sobolev spaces with second time derivative
NOTE: I also asked this question here in MSE.
In the weak theory of evolution PDEs, the Bochner-Sobolev spaces are frequently used. For $a,b \in \mathbb{R}$ and $X,Y$ banach spaces, we define these ...
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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 ...
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Existence of sequence of regular projections
Reading the book :Krasnosel'skii, M.A.; Pustylnik, E.I.; Sobolevskii, P.E.; Zabreiko, P.P. (1976), Integral Operators in Spaces of Summable Functions, Leyden: Noordhoff International Publishing, 520 p....
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Sequential compactness of a sequence of curves of Borel probability measures
$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\DeclareMathOperator*{\supp}{supp}
\newcommand{\bR}{\mathbb{R}}
\newcommand{\bT}{\mathbb{T}}
\newcommand{\bN}{\mathbb{N}}
\newcommand{\bP}{\mathbb{...
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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 $R>0$. ...
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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?
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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,\...
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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\...
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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 ...
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PDE: compactness vs blowup
There are, of course, plenty of approaches on how to solve a non-linear PDE. Two of them are the following:
Solve (easier) approximate problems, show some form of compactness for the approximate ...
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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?
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Density of subsequences in Bolzano-Weierstrass
Let $(M, d)$ be a metric space and $K$ compact. It is known that $K$ is sequentially compact, so we can "run" Bolzano-Weierstrass on it.
I want to identify the set $\mathcal{F}$ of all ...
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finite dimensionality of a subspace of a Banach space
Let $H$ be the space of measurable functions on $(0,1)$ such that
$$ \|u\|_{H}^2 = \int_0^1 x^2\,|\partial_x u|^2\,dx + \int_{0}^1 |u|^2\,dx <\infty.$$
Let $C>0$ be a constant. Suppose that $W \...
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A compact embedding claim
Let $U= (0,1)\times (0,1)$. Consider the weighted Sobolev space $H_1$ with the norm
$$ \|u\|_{H_1}^2 = \int_0^1 \left(\int_0^1 x\,|u(x,y)|^2\,\mathrm{d}x\right) \,\mathrm{d}y.$$
Let $H_2$ be the ...
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Topologies that turn the real numbers into a compact Hausdorff topological group
If I'm not mistaken, the question I put on the title used to be on this site, but I'm not being able to find it at all. I'm therefore reposting it so that someone can either give me the old link or ...
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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,...
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Is bounded-$\mathcal L_{\omega_1,\omega_1}$ significantly different from $\mathcal L_{\omega_1,\omega_1}$?
Take the language $\mathcal L(=,\in)_{\omega_1,\omega_1}$, if we restrict infinite quantification strings, i.e. of the forms $``(\forall v_i)_{i \in \omega}"; ``(\exists v_i)_{i \in \omega}"$, to ...
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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,...
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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 $...
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Convolution of $L^2$ functions
Let $u\in L^2(\mathbb R^n)$: then $u\ast u$ is a bounded continuous function. Let me assume now that $u\ast u$ is compactly supported. Is there anything relevant that could be said on the support of $...
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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 ...
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When is compactness of fiber components an open condition?
Consider a smooth map $f:M\rightarrow N$ between smooth manifolds.
Ehresmann's theorem states that if $f$ is a proper submersion, then it is locally trivializable; in particular, this implies that ...
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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 ...
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Poisson equations for tensors on compact Riemannian manifold
Let $({M},g)$ be a compact Riemannian manifold with Levi-Civita connection $\nabla$. It is well known that the Poisson equation
$$\Delta f=S$$
where $\Delta:C^{\infty}({M})\to C^{\infty}({M})$ denotes ...
user199422