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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Prove compact of a set

Could anyone help me to show that the sets $\{(x,y)|f(x,y)\le \gamma, x>0, y>0\}$ are compact for all scalars $\gamma$, for the function $f(x,y)=xy+\frac{1}{x}+\frac{1}{y}$? I think it is easy ...
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29 views

A question about compactly generated topology

Given a space $X$ and a collection of subspaces $X_\alpha$ whose union is $X$, these subspaces generate a possibly finer topology on $X$ by defining a set $A\subset X$ to be open iff $A\cap X_\alpha$ ...
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1answer
66 views

Relationship between $d(A,B)\gt 0$ and $A \cap B = \varnothing$

a) Show there exists closed non-compact subsets in $\mathbb{R}^2$ such that $d(A,B) = 0$ and $A \cap B = \varnothing$ b) Given $K$ being compact and $B$ closed, show there is a sequence $x_n \in K$ ...
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16 views

compactness or not of a Lie group

Is the Lie group generated by this Lie algebra compact or not? $$ [X_i,X_j]=0, [H_i,H_j]=f^{ijk} X_k, [X_i,H_j]=0 $$ $f^{123}>0$, and $i,j,k \in \{ 1,2,3\}$. There are 6 generators in ...
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1answer
30 views

Is this strengthening of paracompactness known?

Consider a topological space $X$. What can be said about the following property? For any open cover $\mathcal U = \{ U_i \}_{ i \in I }$ of $X$, there exists an open refinement $\mathcal V = \{ V_j ...
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1answer
32 views

True or False statements about compactness of Lie group

Several statements I like to know their True or False statements about the compactness of Lie group. Semi-simple Lie algebra: Every semi-simple Lie group generated by the semi-simple Lie algebra is ...
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3answers
36 views

Prove the intersection of a compact set and a set with no accumulation points is finite

Let $S\subset\mathbb{C}$. We say that $z_0$ is an accumulation point of $S$ if for every $r>0$, the intersection $D(z_0,r)\cap S$ is an infinite set. Let $U\subset\mathbb{C}$ be an open set such ...
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1answer
31 views

In a locally compact Hausdorff space, why are open subsets locally compact?

Let $X$ be a locally compact Hausdorff space, and $A \subset X$ closed. I want to show that $X - A$ is locally compact. I have found a proof here: Open subspaces of locally compact Hausdorff spaces ...
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1answer
16 views

Non-Lipschitz homeomorphism from compact metric space to itself

Is it possible to find a compact metric space $(X,d)$ with more than one point and a homeomorphism $\varphi:(X,\tau) \to (X,\tau)$ where $\tau$ is the topology induced by $d$ such that $$(\forall N\in ...
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25 views

How can I prove that it isn't a compact space [closed]

Let $X=N$ and $B$ is a base for topology $τ(B)$ on $N$ . $B$={φ,{0,1,2,3},{4,5,6,7},{8,9,10,11},........} how can I prove that ($N$,$τ(B)$) is not compact space
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37 views

Elementary proof of compact space = exhaustible space?

(This is a repost of a question I asked last year on cs.stackexchange.) The work of Martín Escardó has demonstrated close parallels between classical topology on one hand and computability on the ...
4
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1answer
65 views

Locally Compact Spaces: Characterizations

For Hausdorff spaces the following are equivalent: Every point admits a compact local base. Every point admits a compact neighborhood. Every point admits a precompact neighborhood. Every point ...
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1answer
25 views

Bound for Integrator Operator

Let $E = L^p(0,1)$ with $1 ≤ p < ∞$. Given $u ∈ E$, set $$Tu(x):=\int_0^x u(t)dt$$ Prove that $T$ is compact on $E$. I would like to use Ascoli-Arzela', but I need to prove: $$|T u(x) − T u(y)| ...
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1answer
23 views

Checking that $C_{0}(X)$ is a vector space

I am trying to prove that $C_0(X)$ is closed subspace of $C_b(X)$ (bounded continuous functions) Given, $X$ is locally compact. $C_0(X)$ is the space of all continuous functions $f:X \to F$ (field of ...
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1answer
42 views

Check if $M = \{z \in \mathbb{C}| z = \frac {1}{n} + \frac {i}{m} \ with \ \ m,n \in \mathbb{Z} \backslash \{ 0 \} \} $ is compact

I want to check, if this set is compact: $M = \{z \in \mathbb{C}| z = \frac {1}{n} + \frac {i}{m} \ with \ \ m,n \in \mathbb{Z} \backslash \{ 0 \} \} $ Thoughts: $z:= a +bi$ real part $a$ is ...
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1answer
19 views

What's the meaning of the state space with locally compact topological space?

I have encountered a statement in one paper describing the continuous-time controlled Markov chain with space state which is locally compact topological space. What does this mean? In my previous ...
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2answers
20 views

Condition that a local homeomorphism be a covering map.

Let be $f:Y\to X$ a local homeomorphism, with $Y$ a compact space and $X$ a Hausdorff connected space. How can I show that, for each $x\in X$, $p^{-1}(x)\subset Y$ is finite? So, is clear that $f$ is ...
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2answers
22 views

Compact subset of space of matrices and compactness verification of a set of eigenvalues

Let $M_n(\mathbb R)$ be the vector space of real matrices of size $n$ , identified with $\mathbb R^{n^2}$ ; let $X \subseteq M_n( \mathbb R)$ be a compact set ; let $S \subseteq \mathbb C$ be the set ...
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1answer
33 views

Countable union of compact sets is compact?

Let $A_0$ be a compact set (closed and totally bounded in some metric space) and consider a sequence of sets $A_n=\{x:d(x,A_0)<1/n\}$. For each $n$, $A_0\subset B_n\subset A_n$ is compact. ...
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1answer
49 views

Uniform Convergence to the Exponential Function over a Compact Interval

I'm trying to show that the sequence of functions $f_n(x)=(1+(x/n))^n$ converges uniformly to $f(x)=e^x$ over any compact interval of the real line. We're assuming that it converges pointwise. Here is ...
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1answer
24 views

James' theorem—going from the separable case to the general case

Consider the following famous theorem by Robert C. James (1964): Let $X$ be a Banach space over $\mathbb R$ and $C$ a non-empty, bounded, weakly closed subset. Then, $C$ is weakly compact if and ...
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1answer
38 views

Metric $p := p(x,y)= \min(|x-y|, 1- |x-y|)$ $x,y \in [0,1)^2$. Prove metric space is compact.

Help! I know that $X$ is Compact if every sequence in $X$ has a subsequence converging to a point in $X$. Also we have that $X$ is a bounded infinite subset in the real numbers. I think it's quite ...
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1answer
50 views

Complement of a point of a Compact Connected Hausdorff Space has no compact maximal connected subspace

This question is a slight modified version of Compact Connected Hausdorff Space has no compact component in the complement of a point Let $X$ be a Hausdorff Compact Connected Space. Prove that ...
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1answer
56 views

Compact Connected Hausdorff Space has no compact component in the complement of a point

Let $X$ be a Hausdorff Compact Connected Space. Prove that $X\setminus\{x\}$ can't be expressed by the disjoint union of two connected sets with one them being compact.(lets assume the empty set is ...
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1answer
23 views

Show compactness of $E\cup S_1$

Consider $$ S_1:=\left\{z\in\mathbb{C}: \lvert z\rvert =1\right\},\\E:=\left\{0\right\}\cup\bigcup_{n\in\mathbb{N}}\left\{(1-2^{-n})e^{\pi i k/2^n}: ...
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1answer
38 views

Check if the given set is Connected and Compact.

$S=\{\dfrac{x^{2}}{1+x^{2}}:x \in \mathbb R\}$ Since $S$ is not closed (the limit point $1$ does not belong to the set), so I concluded that $S$ is not compact. I am confused about verifying ...
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2answers
60 views

Regularity of Dirac measure on Baire sets

Suppose $X$ is a locally compact Hausdorff space. Define the Baire sets in $X$, denoted by $\mathcal Ba(X)$, to be the smallest $\sigma$-algebra that contains all compact $G_\delta$ subsets of $X$. ...
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174 views

non-symmetric version of compact = totally bounded + complete

It is well-known that a metric space is compact iff it is totally bounded and complete. More generally, it is well-known that a uniform space is compact iff it is totally bounded and complete. Is ...
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0answers
35 views

How do we call such a compactification?

Let $E$ be a denumerable set and let $\mathcal{F}$ be a collection of bounded functions. In the reading we had a compactification of $E$ with respect to $\mathcal{F}$, denoted by ...
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31 views

Open sets in locally compact spaces [closed]

Is every open set in a locally compact space an $F_\sigma$ set? Not assuming any separation axioms
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1answer
121 views

Show that $\hat{E}\setminus E$ is homeomorphic to $S^1$

Set $\mathbb{D}=\left\{z\in\mathbb{C}: \lvert z\rvert <1\right\}$ and define $\mathcal{P}\colon\mathbb{D}\times\mathbb{D}\to\mathbb{R}$ by $$ \mathcal{P}(x,y):=\begin{cases}\frac{1-\lvert ...
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3answers
33 views

Can someone formalize 'compactness of set' defined as follows?

My text book says A set K ⊆ R is compact if every sequence in K has a subsequence that converges to a limit that is also in K. Wondering how to formalize this statement, my trial was this. ...
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1answer
30 views

Compactness is independent of the ambient space

In Rudin's PMA book, 2.33 Theorem states that: Suppose $K \subset Y \subset X$. Then $K$ is compact relative to $X$ if and only if $K$ is compact relative to $Y$. I cannot write down the proof as I ...
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Proving compactness in a geometric scenario

Let $C$ be a compact subset of $R^2$. Let $D$ be the set of all pairs of points $(P,Q)$ from $C$, such that the open segment between $P$ and $Q$ is contained in $C$: $$D = \{(P,Q)|P\in C, Q\in C, ...
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2answers
156 views

Continuous function on a compact set with no fixed points

I'm reviewing this problem for my analysis qual. Let $f:X\rightarrow X$ be a continuous mapping from a metric space to itself. Assume $f $ has no fixed points. Prove that, if $X $ is compact, ...
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Compactification: density of a uniform space $X$ in the spectrum of $UC^b(X)$

First, a small motivation: Suppose we are looking for a compactification of uniform spaces, satisfying an universal property similar to the one of the Stone-Čech compactification of a locally compact ...
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1answer
72 views

Question about quotient of a compact Hausdorff space

I am reading the book 'Algebraic Topology' by Tammo Tom Dieck. On page 12 in the proposition 1.4.4 he states that : Let $X$ be a compact Hausdorff space and $f : X \rightarrow Y$ be a ...
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2answers
38 views

Conditions that guarantee the existence of a largest piece

Let $m$ be the area measure on $R^2$. Let $S$ be a nonempty set of measureable subsets of $R^2$ ("pieces"). Define the largest piece in $S$ as: $$\arg \max_{s\in S} m(s)$$ I am looking for ...
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2answers
73 views

What is the one point compactification of the reals?

In several of my questions this theorem has come up. What is the one-point compactification of the reals? Does it have to do with limits and dividing by $0$? I vaguely remember something about a ...
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0answers
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Continuity of two variable function with compactness and supremum norm

Please assist me with the following homework problem: Let $X$ and $Y$ be metric spaces and suppose that $Y$ is compact. Let moreover $f: X \times Y \to R$ be a continuous function, and define ...
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1answer
28 views

Question about Compactly Supported functions

Suppose I have a compactly supported function $f$ defined with $supp f \subset I= ]0,1[$. Let $K=supp f$. Is this statement true: We can assume without loss of generality that there exists $ ...
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2answers
45 views

If $f$ is a continuous function from $R^3$ to $R$ and $K⊂R^3$ is compact, show that there exist two points $a, b ∈ K$ so that $f(K)⊂[f(a),f(b)]$

If $f$ is a continuous function from $R^3$ to $R$ and $K⊂R^3$ is compact, show that there exist two points $a, b ∈ K$ so that $f(K)⊂[f(a),f(b)]$. When is $f(K)=[f(a),f(b)]$? What I believe is the ...
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2answers
52 views

If $T \models \phi$ then there is a finite subtheory $T' \subset T$ such that $T' \models \phi$

Use the Compactness Theorem to show: if $T \models \varphi$ then there is a finite subtheory $T' \subset T$ such that $T' \models \varphi$. I don't see how I can use the compactness theorem here. ...
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1answer
98 views

Why compact-open topology implies joint continuity?

On page 76, A guide to topology by Steven Krantz, there is a motivating question: If $\mathscr E$ is a family of function from $S$ to $\mathbb R$, then under what circumstances is the mapping ...
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1answer
73 views

Compactness and completeness in groups

I know that, in metric spaces, compactness implies completeness. In fact, (i) compactness is equivalent to the fact (ii) every infinite set has an accumulation point and to the fact that (iii) any ...
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Proof of the Riesz-Schauder Theorem (for compact operators) using the Analytical Fredholm Theorem

First of all sorry for my bad English, I'm an Italian student, hope to let you understand! I'm having a little troubles with the proof of the Riesz-Schauder theorem for Compact Operators. Some infos ...
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1answer
34 views

If $X$ is a compact metric space and $f:X \to Y$ is a continuous map , where $Y$ is another metric space , then is $f(X)$ a complete subset of $Y$ ?

If $X$ is a compact metric space and $f:X \to Y$ is a continuous map , where $Y$ is another metric space , then is $f(X)$ a complete subset of $Y$ ?
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1answer
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Looking for counter example - compactness theorem

Let $S$ be a family of sets. We say a subset $S'\subseteq S$ is good if we can choose from every set $A\in S'$ a representative $x_A$ s.t.: For every three sets $A,B,C\in S'$ it holds that $(x_A + ...
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1answer
50 views

Continuous extension on compact set in $\mathbb{R}^n$

I'm an undergrad student reading through Deimling's Nonlinear Functional Analysis and have come across the following proposition. Let $A\subset\mathbb{R}^n$ be compact and $f:A\to\mathbb{R}^n$ be a ...
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1answer
30 views

Is this a compact metric space?

Consider a fixed set of finite discrete symbols $\mathcal{A}$. Equip $\mathcal{A}$ wit the discrete topology which we denote by $\theta$, and $\mathcal{A}^{\mathbb{Z}^d}$ with the product topology, ...