Questions tagged [compactness]
The compactness tag is for questions about compactness and its many variants (e.g. sequential compactness, countable compactness) as well locally compact spaces; compactifications (e.g. one-point, Stone-Čech) and other topics closely related to compactness. This includes logical compactness.
6,386
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Why can't I immerse the unit interval $[0, 1]$ in the 1-torus $T^1$?
I'm supposed to prove that I can't smoothly immerse a compact, simply connected and smooth 1-manifold into $S^1 = T^1$, the 1-torus. I don't understand why I can't do this: take my simply connected 1-...
- 1,070
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Uniformly equicontinuous
Let $X$ and $Y$ be metric space and $a \in X$. A family $A$ of functions from $X$ to $Y$ is said to be equicontinuous at $a$ if for any $\epsilon >0$ there exists a $\delta >0$ such that $d(x,a)&...
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Is the embedding $X_T := \mathbf{H}(\text{curl}) \cap \mathbf{H}_0(\text{div}^0) \rightarrow \mathbf{H}_0(\text{div}^0) $ compact?
I am currently studying embeddings in Sobolev spaces related to computational electromagnetics, and I came across a problem that I'm not entirely sure about.
Specifically, I am interested in the ...
- 79
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$A$ and $B$ is compact space of normed vector space then $A+B$ is compact. [duplicate]
If $A$ and $B$ are compact subset of a normed vector space then $A+ B$ is also compact space, right?
Because since $A\times B$ is compact and $f:A\times B \rightarrow A+B$ such that $f(a,b) = a +b $ ...
- 41
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Finite Boolean combinations of rational intervals, as a subspace of $\{0,1\}^{\mathbb{R}}$: is it $C^*$-embedded?
Setup: Consider $X = \lbrace 0,1\rbrace ^{\mathbb{R}}$ with the product topology, which we might freely identify with the powerset of $\mathbb{R}$ through characteristic functions.
Let $D \subseteq X$ ...
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Does the Space of Functions have a Compact Covering?
Let $X$ be an arbitrary set and $\mathbb{R}^X$ denote the set of real valued functions on $X$. Define the metric $$d(f,g) = \sum_{i=1}^\infty \frac{i \wedge \sup_{x \in X} |f(x) - g(x)|}{2^i}$$ on $\...
- 1,702
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Proof regarding the equivalence of norms, compactness and completeness of normed vector spaces of finite dimension
I wrote a proof regarding the important properties of normed vector spaces that are of finite dimension. I would like to know if I've made mistakes of any sort. I'm open to any critique.
$\mathbb{K}$ ...
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2
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120
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Locally Lipschitz with respect to a variable uniformly to another implies Lipschitz for every compact subset
I was reading a proof of Picard-Lindelöf theorem and there is a step in the proof which needs the following proposition, which I have not been able to prove.
Let $$f:A\subset{\mathbb{R}^{n+1}}\to{\...
- 55
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Can ordinal spaces be Dowker spaces?
Can an ordinal space be Dowker? My guess is no; however, I don't have a proof. Here is my progress:
I know Dowker spaces are not countably paracompact.
It seems that if the ordinal $\alpha$ is a ...
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Is $\Gamma \vDash B$ in propositional logic a r.e. problem?
One version of the compactness theorem in propositional logic is as follows: if $\Gamma \vDash B$, then there is a finite $\Delta \subseteq \Gamma$ s.t. $\Delta \vDash B$.
However, this by itself does ...
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Why do we need step 4. in this proof of the compactness of $[0, 1]$?
In this proof of the compactness of $[0, 1]$ provided in this question
I am understanding proof of theorem stated in title from Spivak's calculus. It is as below.
(0) Let $\mathcal{O}$ be an open ...
- 545
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1
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Jacod and Protters Proof of characteristic uniqueness of functions, extending from compact to infinity.
I want to fill in the details in Jacod and Protter's proof of the uniqueness of characteristic functions. The problem is that we have something that works for a compact set, and I want it to work for ...
- 1,446
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50
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Limits of sequences of nonlinear "bounded" operators
I have a sequence of nonlinear operators $T_n\colon X \to Y$ between two separable Hilbert spaces. The operators are bounded uniformly in that there is a constant $C$ with
$$|T_n(x)| \leq C|x|.$$
I ...
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Understanding the witness property in the Henkin construction
I'm trying to get a really good intuition for why the proof of the compactness theorem via the Henkin construction in Marker's model theory uses the witness property, specifically, why use
For every ...
- 3,172
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What is the map of the Stone-Čech compactification of the rationals to that of the reals like?
This is a kind of followup question to this old one.
Let $\mathbb{Q}$ and $\mathbb{R}$ have their usual (Euclidean) topology, and let $\beta\mathbb{Q}$ and $\beta\mathbb{R}$ stand for their respective ...
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Do additional boundary conditions affect the compact embedding of Sobolev spaces?
Suppose $\Omega \subset \mathbb{R}^2$ open bounded and sufficiently smooth with outward unit normal $\mathbf{n}$.
It is known by the Rellich-Kondrachov theorem that
$$ H^2(\Omega) \hookrightarrow H^1(\...
- 97
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1
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Quasicompactness of $\operatorname{Proj}A$
$\newcommand{\Proj}{\operatorname{Proj}}$
Let $A$ be a graded ring of the form:
\begin{align}
A=\bigoplus_{n=0}^\infty A_n
\end{align}
So in particular this ring is positively graded, but it is not ...
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If $(X,\tau)$ is compact and $\mathcal{F}\subset C(X)$ is separating points, then the weak topology generated by $\mathcal{F}$ is equal to $\tau$
Let $\left(X,\tau\right)$ be a compact topological space, and let $\mathcal{F}\subset C(X)$ such that $\forall x\neq y\in X,\exists f\in\mathcal{F}:f(x)\neq f(y)$. Then the weak topology generated by $...
- 377
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Heine-Borel for Finite Dimensional Normed Vector Spaces
I would like to show that finite-dimensional normed vector spaces have the Heine-Borel property (any subset is compact if and only if it is closed and bounded). I have decided to take the following ...
2
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Compactness criterion in the Skorokhod space $\mathbb D([0,T],E)$ with separable $E$
For a metric space $(E,d)$, we define the Skorokhod space $\mathbb D([0,T],E)$ as being the set of all functions from $[0,T]$ to $E$ which are right-continuous and with left limits.
In the case where $...
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73
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Consequence of Banach Alaoglu Theorem
The Banach Alaoglu theorem states:
Let $\mathcal{Z}$ be a banach space. The closed unit ball
$\{Z^{\ast}\in\mathcal{Z}^{\ast}:\|Z^{\ast}\|_{\ast}\leq1\}$ is compact in the weak$\ast$ topology of $\...
- 537
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votes
1
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88
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Pointwise convergence of almost periodic function
This question is inspired by this post and definitions I use can be found there. The definition of $C_0(\mathbb{R})$ can be found in this wiki:
Fix $f$ a real-valued almost periodic (but not periodic) ...
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1
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If $\{K_n\}_n$ is a family of compacts such that $\bigcap_nK_n=\varnothing$, then $\exists N\in\mathbb{N}$ such that $\bigcap_{i\le N}K_i=\varnothing$
I was reading Bogachev's Measure Theory book, when I stumbled upon this part:
The specification "more generally, in a topological space" made me wonder if the same result can be proved for ...
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249
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Isometry into the Hilbert cube
Let $(X,d)$ be a compact metric space. Does there exist a metric $d'$ on the Hilbert cube $H = [0, 1]^\mathbb{N}$, compatible with its topology, such that $(X,d)\hookrightarrow (H,d')$ is an isometric ...
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1
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Weak* compactness of subset of $L^q$
Let $\mathcal{Z}=L^p(\Omega, \mathcal{A}, \mu)$ and its dual space $\mathcal{Z}^*=L^q(\Omega, \mathcal{A}, \mu)$. Let
$$\langle Z, Z^*\rangle = \int Z \, Z^* d\mu$$
be their dual paar.
EDIT: Let $c:\...
- 537
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What fields generate compact projective spaces?
Given a field $k$, one can generate the vector space $k^n$ and topologize it with the product topology. Then one can delete the 0 element and quotient the remaining set ($k^n \setminus \{0\}$) by the ...
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Compactness of set of all measurable functions
Let $X$ be compact subset of $\mathbb{R}^n$ and let $Y$ be compact subset of $\mathbb{R}^p$. Consider $\mathcal{F}$ the set of all measurable functions from $X$ to $Y$. Can I find topologies under ...
- 537
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Showing One-point compactification topology on $\mathbb R^n$ is Compact without using Stereographic Projection map. [duplicate]
The following is a well known result from point-set topology:
Theorem:
Suppose $\mathbb R^n$ is the Euclidean-$n$-space and $\infty$ be a symbol not contained in $\mathbb R^n$(By Russel's Paradox)....
- 3,693
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Proving $\sigma$-compactness of Pontryagin dual
I am reading A First Course in Harmonic Analysis(2nd ed.) by Anton Deitmar. In Theorem 7.1.4, the author proves that the dual group $\hat{A}$ of a LCA group $A$ is also an LCA group. I am stuck with ...
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Is there a continuous bijective mapping of $R$ into a compactum
I wonder if there are generally no bijective, continuous mappings of the form $$f: \mathbb{R} \rightarrow K$$ if $K$ is a topological compact space.$$$$ My considerations I realize that the inverse ...
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When must a space generated by compacts also be generated by Hausdorff compacts?
Cross-posted to MathOverflow: https://mathoverflow.net/questions/475934/.
I'm interested in comparing $k_1$-spaces,
spaces whose topologies are witnessed by
their compact subspaces, and $k_3$-spaces,
...
- 6,842
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For compact Hausdorff spaces, is countable pseudocharacter equivalent to first countable? [duplicate]
Let $X$ be a compact $T_2$ space. Is $X$ first countable if, and only if, $X$ has countable pseudocharacter?
Note: I have already proven that every $T_1$ first countable space has countable ...
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Rudin's RCA 2.24 theorem : Lusin's theorem
As precised in the title of my question, the context is the book of Walter Rudin : Real and Complex analysis. And especially the proof of theorem 2.24 (Lusin's theorem) which I put below.
I have a ...
3
votes
1
answer
74
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Must scattered spaces with points $G_\delta$ be locally countable?
By M. E. Gewand, “The Lindelöf degree of scattered spaces and their products,” Journal of the Australian Mathematical Society (Series A) 37 (1984), 98–105, we have:
Theorem. Every Lindelöf scattered $...
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Bochner-Sobolev spaces with second time derivative and embeddings
In the weak theory of evolution PDEs, the Bochner-Sobolev spaces are frequently used. For $a,b \in \mathbb{R}$ and $X,Y$ banach spaces, we define these spaces as
$$W^{1,p,q}(a,b,X,Y) = \left\{v, \ \...
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Is there a maximum number of disjoint balls of fixed radius I can fit into a compact metric space?
Let $(X,d)$ be a compact metric space and fix $r>0$. By sequential compactness, one may not find an infinite number of disjoint $r$-balls (sets $B_r(x):=\{y \in X: d(x,y)<r\}$) in $X$ as this ...
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Find compactification of $\mathbb{R}$ which has a subset homeomorphic to $\mathbb{R}^2$
Consider $X=\mathbb{R}$ with the standard topology.
How can I find a compactification $Y$, such that $Y$ is (of course) compact, hausdorff and has a subset which is homeomorphic to $\mathbb{R}^2$?
...
- 3,751
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Locally compact Hausdorff covering space
I am trying to prove the following conjecture:
Let $\pi : E\to B$ be a covering map. If $E$ is locally compact Hausdorff, then so is $B$.
(The converse is known to be true, cf. Exercise 6 in Section ...
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Find a topological manifold that is not Hausdorff and not locally compact
Define a topological manifold as a space locally homeomorphic to $\mathbb{R}^n$. Find a topological manifold that is not Hausdorff and not locally compact. (Hint:Consider $\mathbb{R}$ with "extra ...
- 356
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Uniformly Continuous and locally Lipschitz but not Globally Lipschitz Function on a "Connected but not compact" set
I know such a function exists but I can’t find an example. I have the famous example f:]0,inf[ --->R f(x)=sqrt(x) function. But I can't find any other function which is Uniformly Continuous and ...
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Let $\epsilon>0$ and $f\in \mathscr D(\Omega).$ Then Prove that following sets $K_1$ and $K_2$ are compact and disjoint.
Let $\Omega \subseteq \mathbb R^n$ be an open set.
$\mathscr D(\Omega)=\{f|f \in C^\infty(\Omega) \wedge \text{supp}(f)\subseteq \Omega \text{ is compact.}\}$
Let $\epsilon>0$ and $f\in \mathscr D(...
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Is every compact set equal to an intersection of nested bounded sets with smooth boundary?
Let $K$ be a compact set of (say) the plane $\mathbb{R}^{2}$. Do there exist bounded open sets $(U_{n})_{n\in\mathbb{N}}$ with $C^\infty$-boundary $\partial U_{n}$ such that $$\ldots\subseteq U_{n+1}\...
- 3,396
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1
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75
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Are all complex algebraic varieties in 2+ variables unbounded as a result of Hartogs' extension theorem?
Hartogs' extension theorem states that for any $n\geq 2$, $U\subseteq\mathbb C^n$, $K\subset U$ compact (in $\mathbb C^n$) and a holomorphic function $f:U\setminus K\to\mathbb C$, it can always be ...
- 213
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votes
1
answer
63
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Last Bits of Proof of the Compactness Theorem in Propositional Logic
I am reading the proof of compactness theorem for the propositional logic and the last part of the proof is left as exercise 2 of section 1.7 in the book by Enderton, A Mathematical Introduction to ...
- 15.1k
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0
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79
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Must locally compact and weakly Hausdorff spaces be regular?
In a recent pull request to the π-Base, it was observed that all locally compact and KC (Kompacts are Closed) spaces are regular: since compacts are closed, each local neighborhood base of compacts is ...
- 6,842
4
votes
1
answer
60
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Showing particular quotient space is (not) compact, connected and Hausdorff
I came across the following question, I'm unsure about some of my answers.
Let $U = \{(x, y) \in \mathbb{R}^{2} \mid y \in \{0, 1\}\}$ be a subspace of $\mathbb{R}^{2}$. We define the following ...
- 797
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22
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Understanding the definition of compactly generated space
A compactly generated space (or a $k$-space) is a topological space $X$ where every subset $A\subset X$ is open iff, for every compact subset $K\subset X$, the intersection $A\cap K$ is open in the ...
- 1,682
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0
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27
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Why is sequences enough in the definition of a normal family
In the definition of a normal family in complex analysis, we are concerned with a sequence of functions having a subsequence uniformly converging to a function on compact subsets of an open domain. ...
- 1,682
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1
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47
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Does this theorem hold under specific conditions and not in general?
In baby rudin, Thm 2.34, states:
Compact subsets of metric spaces are closed.
The proof goes as :
Proof Let $K$ be a compact subset of a metric space $X$. We shall prove
that the complement of $K$ ...
- 13
-2
votes
1
answer
53
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Compact set covered by closed balls with positive radii [closed]
Let $K\subseteq\mathbb{R}^2$ be a compact subset. Suppose that it is covered by a collection of closed disks with positive radii. Is there a finite subcover?
- 7,068