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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2answers
22 views

is $\delta$-compact set complete?

We define $\delta$-compact metric space as monotone union of compact sets. $M=\bigcup M_i$ ($M_i\subset M_{i+1}$), is it complete?
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6answers
741 views

Why is an open interval not a compact set?

I learned that every compact set is closed and bounded; and also that an open set is usually not compact. How to show that a concrete open set, for example the interval $(0,1)$, is not compact? I ...
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1answer
28 views

Example of closed unit ball?

I am not understanding the concept of ball on a set $E$ and closed unit ball $B_1$ in $B(E)$. I need to prove or disprove by example that if the closed unit ball $B_1$ is compact or not in a metric ...
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1answer
22 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 ...
2
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3answers
42 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 ...
0
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1answer
35 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 ...
0
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1answer
34 views

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 ...
4
votes
1answer
1k views

Sum of closed and compact set in a TVS

I am trying to prove: $A$ compact, $B$ closed $\Rightarrow A+B = \{a+b | a\in A, b\in B\}$ closed (exercise in Rudin's Functional Analysis), where $A$ and $B$ are subsets of a topological vector space ...
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0answers
38 views

A question about compactly generated topology [on hold]

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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3answers
1k views

Proof of the Compactness Theorem for Propositional Logic

I have a problem understanding the proof for the compactness theorem for propositional logic in my logic course. The compactness theorem states that there is a model for an infinite set $S$ of ...
1
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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 ...
4
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1answer
66 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 ...
3
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1answer
205 views

Stone-Čech compactification. A completely regular topological space is locally compact iff it is open in its Stone-Čech compactification.

I would like to show that a completely regular topological space is locally compact iff it is (weak-star) open in its Stone-Čech compactification. Does this hold in general? I.e given a compact ...
2
votes
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 ...
0
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1answer
17 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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votes
1answer
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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0answers
38 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 ...
6
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1answer
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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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)| ...
1
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1answer
24 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
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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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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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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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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1answer
34 views

Example of Two-point Remainder that are not homeomorphic

We that any two compactification $c_1 N$ and $c_2 N$ of the space $N=D(\aleph_0$) that have finite remainders of the same cardinality are homeomorphic, and yes can be incomparable with respect to the ...
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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 ...
3
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2answers
23 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 ...
0
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1answer
35 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
50 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 ...
4
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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 ...
4
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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
99 views

Existence of a $\sup$ and local compactness in the countable ordinals $\Omega$

I understand how the set of countable ordinals $\Omega$ (with the order topology) is not compact, but how is it locally compact? Also, how can the existence of a least upper bound ($\sup$) for a ...
4
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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
39 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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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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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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0answers
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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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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2answers
391 views

Quasicomponents and components in compact Hausdorff space

Let $X$ be a compact Hausdorff space, $x,y\in X$ and $\mathcal{A}$ a colection of closed subspaces of $X$ such that for every $A\in \mathcal{A}$ then $x$ and $y$ are in the same quasicomponent of $A$. ...
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0answers
16 views

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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0answers
8 views

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 ...
2
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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 ...
0
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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 $ ...
4
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1answer
99 views

Compact subsets of $L^\infty$

The Riesz Frechet Kolmogorov theorem gives a necessary and sufficient condition for a subset of $L^p(\Omega)$ spaces for $1\leq p<\infty$ and equipped with Lebesgue measure to be relatively compact ...
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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
20 views

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 ...