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Questions tagged [compactness]

23 questions from the last 365 days
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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 ...
lulli_'s user avatar
  • 59
2 votes
1 answer
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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_{\...
V. Moretti's user avatar
1 vote
2 answers
202 views

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 $\beta X\setminus X$ ...
Jakobian's user avatar
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2 votes
1 answer
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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 ...
Jakobian's user avatar
  • 1,201
11 votes
2 answers
314 views

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....
Jakobian's user avatar
  • 1,201
1 vote
1 answer
215 views

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 ...
Andrew Luo's user avatar
0 votes
1 answer
145 views

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,...
Brian's user avatar
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7 votes
1 answer
133 views

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. ...
Zhen Lin's user avatar
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1 vote
0 answers
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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 $k_3$-...
Steven Clontz's user avatar
1 vote
1 answer
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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 ...
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3 votes
0 answers
147 views

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 ...
MathsGoose's user avatar
2 votes
1 answer
123 views

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 ...
Ennio's user avatar
  • 23
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0 answers
49 views

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....
Guillermo García Sáez's user avatar
0 votes
1 answer
104 views

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{...
Akira's user avatar
  • 835
0 votes
0 answers
72 views

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 $(...
PDEprobabilist's user avatar
0 votes
0 answers
89 views

Weakly compact set

I want to show that if the set $$ \big\{u \in L^{q}([0, n] ; X): u(t) \in \phi(t, x(t)), t \in[0, n]\big\} $$ is weakly compact, then the set $$ \mathcal{S}_{\phi}(x)=\Big\{u\in L_{loc}^{q}(\mathbb{R}...
Mathlover's user avatar
1 vote
0 answers
75 views

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 $R>0$. ...
Gaspar's user avatar
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1 vote
0 answers
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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?
Peter Kropholler's user avatar
1 vote
0 answers
76 views

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,\...
Gaspar's user avatar
  • 161
5 votes
0 answers
94 views

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\...
Pietro Majer's user avatar
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11 votes
1 answer
308 views

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 ...
Pietro Majer's user avatar
  • 60.5k
3 votes
1 answer
245 views

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 ...
Sebastian Bechtel's user avatar
0 votes
0 answers
63 views

Existence of a measurable maximizer

Let $F$ be a continuous cdf with full support on $[0,1].$ Let $A$ be a compact subset of $\mathbb{R}$ and $\mathcal{M}$ be the set of measurable functions $\alpha:[0,1]\rightarrow A.$ Let $\bar \alpha ...
FeleMath's user avatar