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.

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3
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2answers
81 views
+50

Why if $u_h \rightharpoonup u$ in $*w-W^{1,\infty}$ then exists a subsequence s.t. $u_{h_k}\rightarrow u$ in $L^{\infty}$?

I can't understand why the following fact holds: I consider a sequence $(u_h)\subset W^{1, \infty}(U, \mathbb{R}^N)$, with $U$ open bounded set in $\mathbb{R}^n$, such that $$u_h \rightharpoonup u$$ $...
4
votes
0answers
107 views

On comparing two different notions of compactly generated space

I have encountered, in different circumstances, the following two slightly different categories: The full category of $\mathsf{Top}$ consisting of all objects that are: a) topological spaces ...
2
votes
1answer
35 views

Convergence in compact-open topology implies uniform convergence on compacts

Let $X$ be a topological space and $Y$ a metric space. We give $C(X,Y)$ the compact open topology. If $f_n\to f$ in $C(X,Y)$ then $f_n$ converges uniformly to $f$ on every $K\subseteq X$ compact. ...
-4
votes
0answers
32 views

Lindelöf space with compact space [on hold]

This is one of the exercises from Munkres'. Show that if $X$ is Lindelöf and $Y$ is compact, then $X\times Y$ is Lindelöf.
1
vote
1answer
13 views

Nichtnegativstellensatz the same as Handelman's Theorem?

Wikipedia on "Handelman's theorem: If $K$ is a compact polytope in Euclidean $d$-space, defined by linear inequalities $g_i ≥ 0$, and if $f$ is a polynomial in $d$ variables that is positive on $K$, ...
0
votes
0answers
8 views

compact Lie group with non-compact Lie subgroup? [duplicate]

Can there be compact Lie groups with non-compact subgroups? I thought that was not possible until I thought of the torus with the irrational rotations. So if one identifies $U(1)\times U(1)$ with the ...
0
votes
0answers
28 views

Explicit construction of an $\epsilon$ net covering

Suppose $X$ is a compact space. In particular $X$ is totally bounded and there exists $x_1,..,x_n$ such that $$ X = \bigcup_{i=1}^n U(x_i, \epsilon) $$ where $U$ is the Open Ball centered at $x_i$ ...
1
vote
0answers
39 views

a space isomorphic to $S^{p+q}$

In one of the paper I have met that $$\mathbb{S}^{p+q} \cong \mathbb{S}^p \times \mathbb{R}^q \cup \mathbb{S}^{p-1}$$ I don't know how to deal with $\mathbb{S}^{p-1}$ sphere. Are there any ...
0
votes
1answer
26 views

Construct a global smooth vector field

Assume the following lemma: Let $K$ be a compact subset of a smooth n-dimensional $\mathbb{R}$-manifold $M$ and $U$ an open subset of $M$ such that $K\subset U$. Then there exists a differentiable ...
5
votes
1answer
59 views

An exact sequence of compact topological groups.

Let $A, B, C $ be abelian topological groups such that we have the following exact sequence : $$0\to A \to B \to C \to 0. $$ Assume also that A, C are compact and all the maps are open. Then it's it ...
3
votes
0answers
144 views

Is $X\simeq [0,1]$?

Suppose that $X$ is a metric continuum irreducible between two points $p$ and $q$. Suppose further that whenever $U$ is a connected open set missing $p$ and $q$, we have $X\setminus U$ has two ...
3
votes
1answer
76 views

A set $A \subset l_1$ is compact if and only if closed, bounded, and one other condition

A set $A \subset \ell_1$ is compact if and only if $A$ is closed and bounded and given any $\epsilon >0$, there exists $n_0$ such that $\sum_{k=n}^{\infty} |x_k| < \epsilon$ for all $n> n_0$ ...
6
votes
1answer
245 views

Proving a necessary and sufficient condition for compactness of a subset of $\ell^p$

Let $A \subset \ell^p$, where $1 \le p \lt \infty$. Suppose the following conditions are true: 1) $A$ is closed and bounded 2) $\forall \epsilon \gt 0, \: \exists \: N \in \mathbb{N}$ such ...
0
votes
0answers
2 views

Reference request for "isomorphism upto compact kernel /cokernel “

Let $A$, $B$ be abelian topological groups with a map $f :A \to B$. Assume also that the kernel and cokernel of this map are compact. Then we call f an isomorphism upto compactness. Now let $A, B, C$...
1
vote
1answer
49 views

Show all sequence of $l^1$ with $|x_n|\leq \frac{1}{n^2}$ is compact.

Could you help me to check my proof: let $\{x^k\}$ be a sequence in such set, we use Cantor's diagonal argument to show the existence of convergent subsequence. There exists a subsequence $\{x^{\...
0
votes
2answers
38 views

Let $X$ be a non-empty compact Hausdorff space. Which of the following statements are true?

Let $X$ be a non-empty compact Hausdorff space. Which of the following statements are true? $a.$ If $X$ has at least $n$ distinct points, then the dimension of $C(X)$, the space of continuous real ...
3
votes
1answer
70 views

Show compactness of subset of $\mathbb R^3$

I need to show that $$A:=\{(x,y,z)\in\mathbb R^3; 3x^3y+2xyz^3+2y^2+3=0, xy^3+3xz+x^3=0\}$$ is closed and bounded, hence compact. I don't really know what to do here, can you help?
0
votes
2answers
64 views

$X$ is metric space s.t. for every metric space $Y$ and any continuous function $f : X \to Y$ , $f(X)$ is closed in $Y$ ; is $X$ compact?

Let $X$ be a metric space such that for every metric space $Y$ and any continuous function $f : X \to Y$ , $f(X)$ is closed in $Y$ , then is $X$ compact ? Compare with this $A \subseteq \mathbb R^n $...
-1
votes
1answer
14 views

Partial integration for smooth functions with compact support

Let $d\in\mathbb N$ $\lambda$ be the Lebesgue measure on $\mathbb R^d$ $\Omega\subseteq\mathbb R^d$ be open Why can we use partial integration to obtain $$\int_\Omega\phi\Delta\psi\;{\rm d}\...
1
vote
3answers
44 views

Is the support of a compactly supported function on $\Omega$ a proper subset of $\Omega$?

Let $d\in\mathbb N$ and $\Omega\subseteq\mathbb R^d$ be open. Is there some continuous $\phi:\Omega\to\mathbb R$ with compact support $\operatorname{supp}$ and $\operatorname{supp}\phi=\overline{\...
0
votes
1answer
27 views

If $K'$ is the compact support of a $ℝ$-valued function on an open set $Ω⊆ℝ^d$, can we find a closed ball $K$ with $K⊆Ω$ and $K'⊆K$?

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be non-empty and open $\phi:\Omega\to\mathbb R$ be continuous with compact support $\operatorname{supp}\phi$ Can we find a closed ball $K$ such ...
2
votes
1answer
48 views

$A \subseteq \mathbb R^n $ s.t. for every continuous function $f : A \to \mathbb R$ , $f(A)$ is closed in $\mathbb R$ , is $A$ closed $\mathbb R^n$?

Let $A \subseteq \mathbb R^n $ such that for every continuous function $f : A \to \mathbb R$ , $f(A)$ is closed in $\mathbb R$ ; then I know that $A$ is bounded ; my question is , is $A$ closed in $\...
0
votes
0answers
50 views

How can I prove the following theorem: [duplicate]

Let $\Sigma$ and $\Sigma'$ be satisfiable sets of formulae such that $\Sigma\cup\Sigma'$ is not satisfiable, then there exists a formula $\Phi$ such that $\Sigma\models\Phi$ and $\Sigma'\models\lnot\...
19
votes
2answers
741 views

Topological spaces in which every proper closed subset is compact

Let $X$ be a topological space. It is a basic result that that if $X$ is compact, then every proper closed subset $Y \subset X$ is compact. Out of curiosity, I would like to explore the converse of ...
2
votes
3answers
31 views

Slick proof that continuous image of precompact space is precompact

Is there a slick proof that precompactness is preserved under continuous images? By slick I mean analogous to the following proof that compactness is preserved under continuous images. Is it even ...
46
votes
12answers
4k views

What should be the intuition when working with compactness?

I have a question that may be regarded by many as duplicate since there's a similar one at MathOverflow. The point is that I think I'm not really getting the idea on compactness. I mean, in $\mathbb{R}...
0
votes
2answers
30 views

Prove f(M) is a closed interval given f continuous on M into R, M compact and connected.

I have what I believe is a proof for this question however being how short it is, I have my reservations. Define the function f continuous on M into R. M compact and connected. Since M is compact ...
1
vote
1answer
31 views

Limit point - contains one, or infinitely many points of set?

I'm reading through Functional Analysis by Bachman. He defines a limit point as follows: The point $x$ is said to be a limit point of $A \subset X$ iff for every $r$, $S_r(x) \cap A$ contains ...
0
votes
1answer
24 views

Show the set of solutions to the equation $a_1x_1^m + a_2x_2^m + \dotsb + a_nx_n^m = b$ is compact.

Let $a_1,a_2,\dotsb ,a_n$ be positive real numbers, and $m$ a positive even integer. For a real number $b$, let $S_b$ denote the set of solutions to the equation $a_1x_1^m + a_2x_2^m + \dotsb + a_nx_n^...
0
votes
1answer
37 views

How to prove equivalence of different definitions for compactness?

My workbook considers three different definitions for compactness in logic. It says that it can be shown that these are equivalent, but what would be a strategy to show this? I'm familiar with showing ...
4
votes
4answers
129 views

Characterization of Compact Space via Continuous Function

Let $(X,\mathfrak{T})$ be a topological space. We know that if $X$ is compact and $f:X\to \mathbb{R}$ be any continuous function then $f(X)$ is bounded since the continuous image of a compact set is ...
10
votes
3answers
825 views

M compact $p\in M$ , there exist $f:M-p\to M-p$ continuous bijection but not homeomorphism?

Let M be a compact metric space. We know that if $ g:M\to M$ is a continuous bijection then it's a homeomorphism. But I want to know, if I have a continuous bijection $ f:M - \left\{ p \right\} \to M -...
10
votes
2answers
790 views

Stone-Čech compactifications and limits of sequences

I've been working on some old prelims from my university when they used to just be on point-set topology. We don't cover a couple of the topics so I've been teaching myself some of the material, one ...
-5
votes
0answers
28 views

Stone-Čech compactification of a completely regular topological space

Let $X$ be a completely regular space. show that $X$ is homeomorphic to a dense subset of the commutative $C^*$-algebra $C^b(X)$, and that every function in $C^b(X)$ extends to a continuous function ...
1
vote
1answer
50 views

Proper map iff for every sequence $\{x_{n}\} \to \partial D_{1}$, also $\{f(x_{n})\} \to \partial D_{2}$

We have a proper map $f$, this means a continous map $f \colon X \to Y$, so that for every $K \subset Y$ compact, also $f^{-1}(K)$ is compact. Question: Let $D_{1} \subset R^n$ and $D_{2} \subset R^m$...
0
votes
2answers
51 views

An example of subset $A$ such that $A \cap K$ is open in $K$ for each compact set $K$, but $A$ is not open. [duplicate]

Let $X$ be a topological space. For any $A \subseteq X$, consider two possible conditions on $A$: 1) $A$ is open in $X$; 2) $A \cap K$ is open in $K$, for each compact set $K \subseteq X$. Then $(...
0
votes
1answer
54 views

Graph is closed $\iff$ $f$ is Continuous

Let $X$ be a metric space and $Y$ be a compact metric space. I want to show that the graph of $f$, $G\subset X\times Y$ is closed $\iff$ $f:X\to Y$ is continuous. I'm not sure where the compactness ...
5
votes
1answer
1k views

Graph of continuous function from compact space is compact.

I know this question seems to have been asked hundreds of times, but I don't really see how any of the existing answers address my concern, so I'm hoping that maybe someone here might be able to ...
9
votes
1answer
1k views

$f$ continuous iff $\operatorname{graph}(f)$ is compact

The Problem: Let $(E,\tau_E)$ be a compact space and $(F,\tau_F)$ be a Hausdorff space. Show that a function $f:E\rightarrow F$ is continuous if and only if its graph is compact. My Work: First ...
0
votes
2answers
27 views

Every sequence in a compact set $E$ has at least one accumulation point

Proof by contradiction. Say there is no accumulation point, therefore $$\bigcap_{k \in \mathbb{N}}\Big(\overline{\bigcup_{n \geq k}\{x_n\} }\Big) = \emptyset$$ This is equivalent to $$E = E \ \...
9
votes
2answers
2k views

Examples of compact sets that are infinite dimensional and not bounded

In an infinite dimensional Banach space, does a compact subset have to be finite dimensional? I know it cannot contain any infinite dimensional balls, if this mean it has to be finite dimensional, ...
-1
votes
0answers
20 views

Reference Ascoli-Arzelà Theorem

I am looking for a reference in the literature of the following corollary of Ascoli-Arzelà's theorem : $K\subset\mathbb{R}^n$ is compact. The set $\{f:\mathbb{R_+}\to K \ | \ f \text{ is }1- \text{ ...
14
votes
2answers
501 views

Is a space compact iff it is closed as a subspace of any other space?

I am trying to come up with an alternate definition of a compact topological space that coincides with the usual one. Sorry if my topology is a little rusty. My proposed alternative definition is ...
3
votes
0answers
226 views

Let X be a metric space in which every infinite subset has a limit point. Prove that X is compact.

Let $X$ be a metric space in which every infinite subset has a limit point. Prove that $X$ is compact. The following is my proof I'd like to know if it is correct. Proof: I will use the fact that ...
2
votes
1answer
24 views

Preservation of Compact Sets (confused about counterexample).

This definition for the preservation of compact sets is taken from Abott 2001: Let $f : A \to \Bbb R$ be continuous on $A$. If $K \subseteq A$ is compact, then $f(K)$ is compact as well. I feel ...
1
vote
0answers
17 views

Equivalence between properties of compactness for metric spaces

I am attempting here to show the equivalence between the following three statements for the metric space $(X,d),$ i) $(X,d)$ is compact, meaning every open cover admits a finite subcover ii) $(X,...
4
votes
1answer
79 views

Proof of the Arzelà–Ascoli Theorem

I'm stuck on a particular line of the proof of The Arzelà–Ascoli Theorem. In lectures, we have: $1.$ Defined equicontinuous as: Let $X$ be a metric space, $C(X) = \{f: X \rightarrow \mathbb{R}\...
2
votes
0answers
98 views

The compactness of metric space $\mathcal{G}_n$

Here, metric space $\mathcal{G}_n$ is described below: It is said that it is well-known that $\mathcal{G}_n$ is compact for every $n$. But I can't find a proof, can you give me a proof or an ...
0
votes
1answer
75 views

Is there ever an example of when the union of a space $X$ and a collection of compact subspaces is not compact?

I have seen several examples where the union of a space $X$ and collection of compact subspaces is compact. Is there ever an example of when the union of a space $X$ and a collection of compact ...
0
votes
1answer
26 views

Is $\{(x,y) \in \mathbb R^2 \mid x^2 + x^3y^2 = 0\}$ compact in $(\mathbb R^2, \mathcal E_2)$?

Determine if $X = \{(x,y) \in \mathbb R^2 \mid x^2 + x^3y^2 = 0\}$ is compact in $(\mathbb R^2, \mathcal E_2)$, where $\mathcal E_2$ denotes the standard Euclidean topology. I know that $X$ is ...