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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$X$ is a normed linear space such that for some compact $K\subseteq X$ , $\operatorname{span} K$ is dense in $X$ then is $X$ separable?

Let $X$ be a normed linear space which is separable. Then I know that there exists a compact subset $K$ of $X$ such that $\operatorname{span} K$ is dense in $X$ (in fact we can also find compact and ...
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31 views

If $X$ is a compact metric space and $E_n$ is closed nonempty subset, show that $\cap_{n=1}^\infty E_n$ is nonempty.

Suppose that $(X,d)$ is a compact metric space and $(E_n)$ is any sequence of nonempty closed subsets of $X$ with $E_{n+1}\subset E_n$ for all $n\in\mathbb{N}$. Show that $\cap_{n=1}^\infty E_n$ is ...
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25 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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1answer
15 views

Nonlinear operator sends bounded set to relatively compact set

Consider $g$ a continuous function on $[a,b]\times\mathbb{R}$, and let $z_0\in\mathbb{R}$. Define the (nonlinear) operator on $C[a,b]$: $$Mv(x)=z_0+\int_a^x g(t,v(t))\,dt$$ for $x\in[a,b]$. Prove ...
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27 views

Compact set of a set $K$

Show that $K := \{x \in C[0,1] : x(0) \in [-3,4], |x(t)-x(s)| \leq d |t^2-s^2|, \forall t,s \in C[0,1]\}$ is compact. I already know that $C[0,1]$ is a compact set. So it is only necessairy to show ...
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1answer
36 views

Show that $f$ is uniformly continuous - Compactness

Let $K := \{x \in C[0,1] : x(0) \in [-3,4], |x(t)-x(s)| \leq d |t^2-s^2|, \forall t,s \in C[0,1]\}$. Let $y \in C[0,1]$ and $f : K \to \mathbb{R}$ defined as $f(x)= \int_0^1 x(t)y(t)dt$. Show ...
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17 views

Show that $Gr(f)$ is compact

Let $A \subset \mathbb{R}^n$ a compact and $f : A \to \mathbb{R}^m$ a continuous function. Let the graph of $f$ $$Gr(f) = \{(x,f(x) : x \in A)\}.$$ Show that $Gr(f)$ is compact. My proof : ...
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1answer
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Help understanding a particular proof of the compactness theorem for Propositional Calculus.

I've reading through this proof, I don't understand the last part: the claim $\tau \models \Sigma$. Note: I'll use $AP(\varphi)$ and $\text{Var}(\varphi)$ interchangeably, to mean the variables that ...
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27 views

Unbounded, closed, star-shaped set contains ray

I am trying to prove the following statement: Let $R$ be a real closed field (such as the real numbers). Let $M\subseteq R^n$ be a semi-algebraic set, i.e. a set which is defined by a Boolean ...
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1answer
19 views

Showing uniform continuity for two intervals

Show that if $f$ is continuous on $[0, \infty)$ and uniformly continuous $[a, \infty)$ for some positive constant $a$, then $f$ is uniformly continuous on $[0, \infty)$. Here is my attempt at the ...
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1answer
24 views

Why is it that for any rational numbers $a < b$, the interval $[a, b]$ in $\mathbb{Q}$ is not compact with respect to this metric?

Suppose $q$ is any nonzero rational number and $p$ is a fixed prime. If $q = p^k\frac{n}{m}$ for integers $n$ and $m$, neither of which has $p$ as a factor, then we define $|q|_p := p^{−k}$. We can ...
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51 views

Is every compact metric space finite dimensional?

It certainly is true if the metric space is normed by Riesz lemma.
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Proof that space of correlation matrices is compact

An $n\times n$ real symmetric matrix is a correlation matrix, if it is positive-semidefinite and all its diagonal entries equal 1. According to most references it is easy to see that the space of ...
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2answers
40 views

Many point compactification

If $X$ is a noncompact LCH space (locally compact, Hausdorff) then its one point compactification is $X^*=X\cup \{\infty\}$ with topology $\mathcal{T^*}$ given by $U \in \mathcal{T^*}$ iff either a) ...
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1answer
19 views

Show that every open set in second countable LCH space is $\sigma$-compact

Let $(X,\tau)$ be a second countable, locally compact Hausdorff space. Theorem: If $S \in \tau$, then $S$ is $\sigma$-compact. How do I show this statement? The following is what I have ...
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1answer
29 views

Show that $d(E,F) > 0$ - Is $E \times F$ is a compact set here?

Let $d(E,F)=\inf\{|z-w| : z \in E, w \in F\}$ where $E \subset \mathbb{C}$ is compact and $F \subset \mathbb{C}$ is closed such that $E \cap F = \emptyset$. Show that $d(E,F) > 0$; namely, ...
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1answer
25 views

Cantor's Intersection Theorem with closed sets [duplicate]

Cantor's Intersection Theorem states that "if $\{C_k\}$ is a sequence of non-empty, closed and bounded sets satisfying $C_1 \supset C_2 \supset C_3 \dots$, then $\bigcap_{n \ge 1} C_n$ is nonempty. ...
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42 views

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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1answer
34 views

Continuous proper map into compactly generated Hausdorff space is closed

Throughout, 'proper' means 'pulls back compacts to compacts'. I've read here and there some claims about properness implying closedness. I want to check whether my attempt at a generalization is ...
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56 views

Is every compact space locally compact?

Suppose that $(X,\tau)$ is a topological space. If $(X,\tau)$ is compact, then $(X,\tau)$ is locally compact. Does this statement hold for any $(X,\tau)$, or does it only hold when $(X,\tau)$ is ...
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Trouble with exercise from Lee's Introduction to Smooth Manifolds

Maybe I'm missing something, but this seems like a mistake. Exercise 2.9 (page 36) says: Show that an open cover $\{U_\alpha\}_{\alpha\in A}$ is locally finite if and only if each $U_\alpha$ ...
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Is the set $\{x \in \mathbb{R}^3 : |x_1-2| + |x_2-1| \leq 2\}$ compact?

Is the set $\{x \in \mathbb{R}^3 : |x_1-2| + |x_2-1| \leq 2\}$ is compact? I know that if $x \in \mathbb{R^2}$, then the previous set would be compact. However, there is an extension, and the ...
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1answer
34 views

Compactification of Lie Group

Is there a way to embed a Lie Group $G$ into a compact lie Group $H$, such that the inclusion is a Lie group homomorphism?
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44 views

Product of compacts is compacts using closedness of projection to second component?

A well known characterization of compactness says $X$ is compact iff for all spaces $Y$ the projection $X\times Y\rightarrow Y$ is a closed map. I'm wondering whether there's some simple formal way ...
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23 views

Compactness of the outer measure

I have maybe a naive question about compactness of outer measure (or completion). Let $(E,\mathcal B(E))$ be a Polish space, and $\mathcal M_b(E)$ the bounded Radon measure on $E$. Assume that a ...
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33 views

Pre-compactness in $L \log L$

As far as I know Zygmund class of Orlicz spaces or "$L \log L$" is defined as an Orlicz space with the Young function $Q(t) = t \sqrt{\ln(1+t)}$ (or something similar to this in different ...
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3answers
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Is the set $M=\{[x,y,z]\in{\mathbf R}^3 :\ x^2 + y^2 +z^2 + xy + yz + xz = 1,\ x \ge 0,\ y \ge 0\}$ compact?

How do I show that the set $$ M=\{[x,y,z]\in{\mathbf R}^3 :\ x^2 + y^2 +z^2 + xy + yz + xz = 1,\ x \ge 0,\ y \ge 0\} $$ is compact? The set is apparently closed (as it cannot be open given that the ...
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1answer
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A 'weird' topology

I've got some questions about the following situation, and some proof-verification requests. Let $\mathcal{T}$ be the smalles topology on $\mathbb{R}$ with the property that ...
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proof of a convergent subrow in every row in B using diagonal argument

full question I got a hint that for the compactness of M I need to show that every row in B has a convergent subrow (diagonal argument) but I don't know how to show this, does anybody know how to ...
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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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1answer
25 views

Can every continuous function be extended from a locally compact Hausdorff space to the one-point compactification?

In my book it is discussed continous functions on a locally compact space in $\mathbb{R}^d$. And they talk about functions that vanish at infinity. When saying this are then they talking about the ...
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37 views

Why to define compactness with sequence?

We defined compactness by can be covered by finite open sets, then we introduced a sequence definition. Why to define compactness again with sequence? What's the use of sequentially compactness?
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Why is $[-1,1]$ compact when $a_n = (-1)^k$ does not converge in $A$

I know this question sounds silly but I was reading the definition of compactness and couldn't quite wrap my head around this Compactness :A subset $A$ of a metric space $M$ is compact if every ...
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1answer
43 views

Verifying if a set is a compact set

I have the following sets and I need to determine which are compact: $$ A = \{(x,y)| x^{2}+ y^{3} \leqslant 10\} $$ $$ B = \{(x,y)| x^{2}+ y^{4} \leqslant 10\} $$ $$ C = \{|x^{2} - 3y|: |x| \leqslant ...
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68 views

Is the set of all diagonalizable matrices compact?

Is the set of all diagonalizable matrices compact? I have got this example \begin{bmatrix} 1 && n \\ 0 && 2\end{bmatrix} which is diagonalizable but not compact because it is not ...
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48 views

Every open set in $\mathbb{R}^n$ is the increasing union of compact sets. [closed]

$X= \bigcup K_m$, where $K_m$ is a increasing sequence of compact sets and $X$ is the open set.
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Criterion for a signed measure to be positive

I am reading a proof in which I do not understand the following claim: Let $K$ be a compact metric space and let $M(K)$ be the set of signed Borel measures on $K$. Then the set $P(K)$ of positive ...
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1answer
20 views

Example of a compact, Kolmogorov space with a basis of compact open sets, but not spectral.

The definition of a spectral space requires four conditions: The space is compact, The space is Kolmogorov (or $T_0$), The compact open subsets form a basis of the topology and are closed under ...
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363 views

$\{ (x,y) \in R^2 \mid x^2 + y^2 -2x + 4y - 11 = 0 \}$ is closed and bounded

As a part of an exercise It would help me if I could prove formally that the set $\{ (x,y) \in R^2 \mid x^2 + y^2 -2x + 4y - 11 = 0 \}$ is closed and bounded. Plotting it with a software I can see ...
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1answer
38 views

I want to prove that $f$ is continuous if its graph is closed

This is an exercise from Rudin's 'Functional Analysis': Suppose that $X$ and $K$ are metric spaces, that $K$ is compact, and that the graph $f:X\to K$ is a closed subset of $X\times K$. Prove that ...
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1answer
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A topological space is compact iff each infinite subset has a complete accumulation point

This is based on the comments and answers provided in this post. However, I have some questions on the proof and the hint given in Kelleys book p.163. I will highlight the hint of the book. My own ...
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1answer
19 views

Showing some spaces are locally compact and show their 1-point compactification is homeomorphic

Consider \begin{align} X_1 &= \{(x,y,z)\in\mathbb{R}^3:(z=0) \text{ or } (x=y=0,z\geq 0)\}, \\ X_2 &= \{(x,y,z)\in\mathbb{R}^3:(z=0) \text{ or } (x=0, y^2+z^2=1, z\geq 0)\}, \\ X_3 &= ...
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2answers
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If the Continuous image of a space is compact, does that mean the space is compact? [closed]

Let $f$ be a continuous function and $f(X)$ is compact. Is $X$ necessarily compact? Is there an example to prove/disprove this? Thank you.
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Continuous functions and open covers

Let $\{V_{i}\}$ be an open cover of $f(X)$. Why is it that if $f$ is continuous then $f^{-1}(V_{i})$ is open? What does the continuity of $f$ have to do with $f^{-1}(V_{i})$ been open?
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Contour Integral Inequality and Normal Families

I am trying to prove a certain family of functions $\mathcal{F}$ is normal, and my proof got very stuck. I am trying to show that the family of analytic functions on $\mathbb{D}$, continuous on the ...
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1answer
66 views

Does there exist a Hausdorff group which is not locally compact?

A topological space is countably compact if every countable open cover has a finite subcover. A topological space $X$ is locally compact if any point has a neighbourhood which is compact. A ...
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1answer
56 views

Show that $S$ is compact.

Let $A \in M_n(\bf C)$ and let $S=\{ A \in M_n(\bf C) : A=A ^*$ and $\rho (A) \leq 1 \}$, $\rho(A)$ denotes spectral radius of $A$. Show that $S$ is compact subset of $M_n(\bf C) \cong \bf ...
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51 views

Proof on compact and closed sets

Take $S$ to be compact and $T$ to be a closed set of real numbers. If we assume that $S \cap T = \emptyset$, I want to prove that there is a $\delta > 0$ such that $|s - t| > \delta$ for every ...
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39 views

Finite sub cover

In the definition of compact set: From any open cover $E=\cup_{i\in I} U_i$ we can find a finite sub-cover $E=\cup_{k=1}^NU_{i_k}$? Is the finite sub-cover is always open please? Thank you.
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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 ...