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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defining a topology by its compact sets

The goal. Let $X$ be a set endowed with Hausdorff topologies $\tau_w$ and $\tau_n$, such that $\tau_w\subseteq\tau_n$. Let $\mathscr{C}$ denote a family of subsets $A\subseteq X$, which satisfies ...
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123 views

An example of a compact multiplicatively unbounded ring

My teacher asked me to build an associative topological Hausdorff compact ring $R$ with $1$, which is multiplicatively unbounded. That means there is a neighborhood $U\ni 1$ such that $FU\not=R$ for ...
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77 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 ...
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148 views

Is dependent choice necessary to prove every perfect compact Hausdorff space is uncountable?

The answer to Cardinality of a locally compact space without isolated point shows that AC is required to show that if $X$ is a compact Hausdorff space with no isolated points then $|X| \ge ...
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38 views

Group orderable iff all its finitely-generated subgroups are orderable

I want to proof this specifically using the Compactness Theorem from propositional logic (this is an exercise from Model Theory, Hodges). $G$ orderable means there is a total ordering s. t. $g\leq h$ ...
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51 views

$E$ compact, real-valued $f : E \to \mathbb{R}$ continuous iff graph is compact - is real valued necessary?

Problem The graph $G$ of $f$ is defined as the points $(x, f(x))$ for $x \in E$. Suppose $E \subset \mathbb{R}$ is compact, then $f : E \to \mathbb{R}$ is continuous iff its graph is compact. ...
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141 views

Compact family of Lip functions under the sup norm metric, proof verification.

Hi everyone I'd like to know if the following is correct, I'd appreciate your opinion and also any suggestion to improve my argument. Thanks in advance for your time. If $(K,d)$ is a compact ...
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81 views

Compactness of a set of bounded functions in the uniform norm

Let $T$ be a non-degenerate compact interval in $\mathbb R$ and $f:\mathbb R^2\to\mathbb R$ a strictly concave function such that (a) $f(0,0)=0$, (b) $f$ strictly increases in the first argument, and ...
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155 views

Locally connected and compact Hausdorff space invariant of continuous mappings

I am looking for a reference (not proof) to the following theorem: If $X$ is a compact and locally connected topological space, Y is a Hausdorff topological space, $f:X\to Y$ is continuous and ...
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87 views

Is $X$ pseudocompact

The following example with a little modified from the handbook of set theoretic topology, Page 574: Let $\kappa$ be any cardinal for which there exists a family $\{H_\alpha: \alpha < \kappa\}$ ...
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286 views

If $f: X \to Y$, when do we have $\beta Y \supset \overline{f(X)} = \beta X$?

Suppose that $X$ and $Y$ are Tychonoff spaces, denote by $\beta X$ and $\beta Y$ their Stone-Čech compactifications and let $f:X\to Y$ be a continuous map. Using the embedding $Y\hookrightarrow\beta ...
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72 views

A question on countably compact space under CH

The question is also posted here. A regular space $X$ is star compact (which implies pseudocompact) with $G_\delta$-diagonal star countable first countable $e(X)\le \aleph_0$ ( in fact it implies ...
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230 views

Constructing the support of a Borel measure

From Rudin, Real and Complex Analysis, Chapter 8, Problem 7, 1st Edition. Suppose $E$ is a compact set in $\mathbb{R}^{k}$ without isolated points. Show that $E$ is the support of a continuous ...
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68 views

Classes of compact spaces

Given a class $\mathcal{C}$ of compact Hausdorff spaces which is closed under countable products and continuous images. Let $\kappa>\omega$ be a cardinal number. Consider the class ...
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79 views

Show precompactness for not necessarily continuous functions

I know that for showing precompactness of subsets of continuous functions, Arzelà-Ascoli is the tool of choice. However, the setting I'm facing here are functions in $L^1$. More precisely: Let ...
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35 views

Rellich-Kondrachov compacteness theorem for the Euclidean space with Gaussian measure

Let $\gamma_n: \mathbb{R}^n\to\mathbb{R}$ be the Gaussian distribution function defined by $$ \gamma_n(x):=(2 \pi)^{-\frac{n}{2}} e^{-\frac{|x|^2}{2}}. $$ Let $d\gamma_n$ denote the following measure ...
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33 views

Regarding Feynman Integration - unnamed theorem (?)

I found a certain theorem, that hasn't been proven in the article and as it seems to require just Calculus 2 (multivariable calculus) knowledge.- I would like to prove it, however I am completely ...
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58 views

Is the Alexandroff double circle compact and Hausdorff?

I recently encountered the Alexandroff double circle. The underlying set is $C = C_1 \cup C_2$, where $C_i$ is the circle of radius $i$ and centre $0$ in the complex plane. The basic open sets are: ...
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64 views

Show that $\varphi : L \to \Bbb{R}$ is continuous.

Let $L,K$ be to compact metric spaces, let $f:K\times L \to \Bbb{R}$ be a continuous function. Define $\varphi : L \to \Bbb{R}$ as $\varphi(y)=\sup_{x\in K} f(x,y)$. Show that $\varphi$ is ...
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57 views

If a set is Hausdorff relative to one topology, can it be compact relative to a strictly finer topology?

Let $\tau_1$ and $\tau_2$ be two topologies on set $X\neq\phi$ such that $(X, \tau_1)$ is Hausdorff and $\tau_1 \subsetneq \tau_2$. Can $(X, \tau_2)$ be compact? My effort: Suppose that $(X, ...
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55 views

Prob. 1, Sec. 27 in Munkres' TOPOLOGY, 2nd ed: How to show that the compactness of every closed interval implies the least upper bound property?

Let $X$ be an ordered set in which every closed interval is compact. Then $X$ has the least upper bound property. How to prove this? My effort: Let $A$ be a non-empty subset of $X$ such that $A$ is ...
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26 views

Does having a real valued cauchy sequence on a function in a compact space imply the function is continous on that space?

I had to prove for a homework assignment this function $$ s_n(x) = \sum_{i=0}^n (-1)^i \frac{ x^{2i+1}}{(2i+1)!} $$ is a Cauchy sequence with respect to the sup norm for $$ s_n : [-M,M] ...
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93 views

When does pointwise convergence on compact space imply uniform convergence?

I just wondered whether there is a more general theorem behind claims like 'if a sequence of equicontinuos functions $f_i:[a,b]\rightarrow{\bf R}$ converges pointwise to a continuous function $f$ then ...
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67 views

Proving amalgamation property in model theory

Restate the proposition Suppose $\mathcal{M}_0$, $\mathcal{M}_1$, and $\mathcal{M}_2$ are $\mathcal{L}$-structures and $j_i ~:~ \mathcal{M}_0 \rightarrow \mathcal{M}_i, ~(i = 1,2)$ is an elementary ...
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298 views

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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114 views

Approximate the unit ball in an infinite-dimensional Hilbert space, by compact sets?

Are there some common ways to approximate the unit ball in an infinite-dimensional Hilbert space, by compact sets? (note that the unit ball isn't compact.) My goal is to prove a statement which holds ...
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27 views

Compactness of solution space of semi-linear parabolic PDE

Under what conditions a closed and bounded subset of solution space of following parabolic PDE is compact? $$x_{t}=x_{zz}+f(x,z)$$ Thank you!
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89 views

Compactness and Lipschitz functions

I am very stumped by this question: Suppose (K, d) is a compact metric space. Let f be any function, f: K $\rightarrow \mathbb{C}$, not necessarily continuous. Prove that for any $\epsilon > ...
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Proof of uniform continuity on compact sets

Show that a function $f:\mathbb{R} \rightarrow \mathbb{R}$ that is continuous on a compact set $K$ is uniformly continuous on $K$. Is the proof below correct? Proof: Let $\epsilon > 0$ and let ...
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163 views

Prove that every pseudocompact metric space is compact

This is from Real Mathematical Analysis by Pugh, problem 2.85(a). I've seen proofs but they've used concepts that haven't been covered up to this point, like the Tietze extension theorem, metrizable ...
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81 views

References for the Čech-Stone compactification of Hyper-Reals?

It seems like $\beta\mathbb R$ has been heavily studied, but I am interested in learning more about $\beta(\mathbb R ^\omega /u)$. $\mathbb R ^\omega /u$ is a proper extension of $\mathbb R$ when ...
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91 views

Lie groups with structure constant $f_{abc} \neq f_{bca}$.

The structure constant $f_{abc}$ of Lie group is defined by the commutators of generators, $$[T^a,T^b]=i f_{abc}T_c$$ automatically $f_{abc}=-f_{bac}$. Can someone give a list of explicit examples ...
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529 views

What is compensated compactness?

As the title says, what is compensated compactness? I see people talk about it in the books and papers I am reading but I can only find hand wavy definitions when I look online. Is there a definition ...
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158 views

Prokhorov theorem in locally compact Hausdorff space?

Prokhorov theorem gives a compactness condition in the space of probability measures on a Polish space. I am wondering whether we have similar conditions for probability measures on more general ...
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180 views

Is this proof correct about compact sets inside open sets?

I've been solving the following problem: "If $U\subset\mathbb{R}^n$ is open and $C\subset U$ is compact, show that there is a compact set $D$ such that $C\subset \operatorname{int}(D)$ and $D\subset ...
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326 views

Every Borel set is the union of an increasing sequence of Bounded Borel sets?

I am currently working with the book by Halmos, and i can't quite get past this one. It states that: "Every Borel set can be written as an increasing sequence of Bounded Borel sets" In this case $X$ ...
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45 views

Strongly additive open cover

Call an open cover $\mathscr{U}$ of a metric space $M$ strongly additive if whenever $U,V\in\mathscr{U}$ and $U\cap V\ne\emptyset$, then $U\cup V\in\mathscr{U}$. Prove that $M$ is compact and ...
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72 views

A question on pseudocompact space

The exercise is from the handbook of set-theoretic topology page 161: Assume $\mathfrak b=\mathfrak c$. Construct a first countable separable zero-dimensional locally compact pseudocompact space ...
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50 views

Another question in relation to Tychonoff theorem

Let $X_i$ be compact topological spaces and let $X = \prod_{i \in I}X_i$ and let $\mathscr F$ be ultrafilter on $X$. Define $\mathscr F_i = \{Y \subseteq X_i : \pi_i^{-1}Y \in \mathscr F\}$. Here ...
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91 views

What is the point of compactness and how does it look visually?

I just learned what the term compact means, as in compact sets. I know what the definition of compactness is, but I don't get what the real significance of it is. How should I try to think of it in a ...
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360 views

Sequential compactness vs. countable compactness

Firstly, I'll give the definitions of sequential compactness and countable compactness. Sequential compactness: If $X$ is a Hausdorff space and every sequence of points of $X$ has a convergent ...
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Interior of a Minkowski sum satisfies Brunn-Minkowski inequality.

If $A,B$ are Lebesgue measurable sets in $\mathbb{R}^n$ with $0 <\lambda(A),\lambda(B)< \infty$. Prove that $\lambda (\text{Int}(A+B))^{1/n} \ge \lambda(A)^{1/n}+ \lambda(B)^{1/n}$ where ...
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38 views

Proper maps in terms of projection from pullback

I've read that a continuous $f:X\rightarrow S$ is proper (inverse images of compacts are compacts) iff for all other continuous maps $g:Y\rightarrow S$ the projection $X\times _SY\rightarrow Y$ in the ...
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45 views

Compactness related property of topological spaces

One of the standard definitions of a compact topological space $\langle X,\mathscr{O}\rangle$ says that $X$ is compact iff every open cover of $X$ has a finite subcover. I would like to ask you if any ...
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118 views

Compactness in $\mathbb{R}^\infty$. (Compact ellipsoid)

Need some help with this question in my Real Analysis course. Take positive decreasing numbers $l_1 \ge l_2 \ge \cdots\ge l_n \rightarrow l_\infty \ge0. $ It is to be shown that the ellipsoid ...
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56 views

Image of a precompact under the action of uniformly continuous function is a precompact

Suppose we have two metric spaces $(X, \rho_x)$ and $(Y, \rho_y)$ and a uniformly continuous function $f\colon X \to Y$. The problem is to prove that image $f(A)$ of every precompact $A \subset X$ ...
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27 views

Compact Subset of a Domain in $\mathbb{C}$

Let $D$ be a domain (an open, connected set) and suppose $K \subset D$ is compact. Now, define for $\eta > 0$ "small enough" the following set: $K_{\eta}=\{z \in D: \mathrm{dist}(z,K) \leq \eta ...
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26 views

Set cannot be Hausdorff or compact with a non-subspace topology

$\mathcal{T}$ is the standard topology and $\mathcal{T}'$ is any other different topology, both on the unit interval $[0, 1]$. Then if $\mathcal{T}' \subsetneq \mathcal{T}$, $[0,1]$ with ...
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25 views

Example of compactness James' Theorem

I would like to find any example on any known space of application of the classical compactness James' Theorem for obtaining a good visualitation of how this result works. For instance, on any space ...
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80 views

Prove that the set of extreme points of a compact convex set is not empty.

The Krein–Milman theorem states that if $S$ is convex and compact in a locally convex space, then $S$ is the closed convex hull of its extreme points. In particular, such a set has extreme points. Is ...