Questions tagged [compactness]
The compactness tag has no usage guidance.
23 questions from the last 365 days
5
votes
1
answer
206
views
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 ...
2
votes
1
answer
156
views
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_{\...
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 $\beta X\setminus X$ ...
2
votes
1
answer
103
views
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 ...
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....
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 ...
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,...
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. ...
1
vote
0
answers
36
views
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 $k_3$-...
1
vote
1
answer
119
views
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 ...
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 ...
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 ...
0
votes
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....
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{...
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 $(...
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}...
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 $R>0$. ...
1
vote
0
answers
127
views
Extremally disconnected sets as building blocks for compact Hausdorff spaces
Is every compact Hausdorff space the filtered colimit of compact extremally disconnected spaces?
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,\...
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\...
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 ...
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 ...
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 ...