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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12
votes
4answers
6k views

Projection map being a closed map

Let $\pi: X \times Y \to X$ be a projection map where $Y$ is compact. Prove that $\pi$ is a closed map. First I would like to see a proof of this claim. I want to know that here why compactness is ...
8
votes
5answers
1k views

How to show that this set is compact in $\ell^2$

Let $(a_n)_{n}\in\ell^2:=\ell^2(\mathbb{R})$ be a fixed sequence. Consider the subspace $$C=\{(x_n)_{n}\in\ell^2 : |x_n|\le a_n\text{ for all }n\in\mathbb{N}\}.$$ According to the book [Dunford and ...
31
votes
5answers
6k views

What's going on with “compact implies sequentially compact”?

I've seen both counterexamples and proofs to "compact implies sequentially compact", and I'm not sure what's going on. Apparently there are compact spaces which are not sequentially compact; quick ...
24
votes
6answers
2k views

What should be the intuition when working with compactness?

I have a question that may be regarded as 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 ...
21
votes
1answer
2k views

Theorem of Arzelà-Ascoli

The more general version of this theorem in Munkres' 'Topology' (p. 290 - 2nd edition) states that Given a locally compact Hausdorff space $X$ and a metric space $(Y,d)$; a family $\mathcal F$ of ...
15
votes
2answers
1k views

If every real-valued continuous function is bounded on $X$ (metric space), then $X$ is compact.

Let $X$ be a metric space. Prove that if every continuous function $f: X \rightarrow \mathbb{R}$ is bounded, then $X$ is compact. This has been asked before, but all the answers I have seen prove the ...
13
votes
5answers
3k views

A compact Hausdorff space that is not metrizable

Is there an example of a compact Hausdorff space that is not metrizable? I was thinking maybe the space of continuous functions $f: X \rightarrow Y$ between topological spaces $X, Y$, might work, but ...
15
votes
2answers
1k views

Isometry in compact metric spaces

Why is the following true? If $(X,d)$ is a compact metric space and $f: X \rightarrow X$ is non-expansive (i.e $d(f(x),f(y)) \leq d(x,y)$) and surjective then $f$ is an isometry.
14
votes
8answers
2k views

How to understand compactness? [duplicate]

How to understand the compactness in topology space in intuitive way?
6
votes
2answers
2k views

Compactness in the weak* topology

Let $X$ be a Banach space, and let $X^*$ denote its continuous dual space. Under the weak* topology, do compactness and sequential compactness coincide? That is, is a subset of $X^*$ weakly* ...
10
votes
2answers
985 views

compactness / sequentially compact

I'm looking for two examples: A space which is compact but not sequentially compact A space which is sequentially compact but not compact Explanations why the spaces are compact / not compact and ...
2
votes
3answers
1k views

Is a closed subset of a compact set (which is a subset of a metric space $M$) compact?

Is there a way to prove this using sequential compactness instead of open cover definitions? My first gut reaction was that the fact was obvious since we can show that the closed subset $[a,b]$ is ...
11
votes
6answers
689 views

Pseudocompactness does not imply compactness

It is well known that compactness implies pseudocompactness; this follows from the Heine–Borel theorem. I know that the converse does not hold, but what is a counterexample? (A pseudocompact space is ...
1
vote
1answer
94 views

Assume that $(\text{X}, T)$ is compact and Hausdorff. Prove that a comparable but different topological space $(\text{X},T')$ is not.

Say that a topological space is CH if it is both compact and Hausdorff. Let $T$ and $T'$ be two topologies on the same set X that are comparable but different, i.e., $T$ is either strictly ...
74
votes
12answers
6k views

Why is compactness so important?

I've read many times that 'compactness' is such an extremely important and useful concept, though it's still not very apparent why. The only theorems I've seen concerning it are the Heine-Borel ...
19
votes
10answers
5k views

How to prove $[a,b]$ is compact?

Let $[a,b]\subseteq \mathbb R$. As we know, it is compact. This is a very important result. However, the proof for the result may be not familar to us. Here I want to collect the ways to prove $[a,b]$ ...
13
votes
3answers
1k views

Countable compact spaces as ordinals

I heard at some point (without seeing a proof) that every countable, compact space $X$ is homeomorphic to a countable successor ordinal with the usual order topology. Is this true? Perhaps someone can ...
14
votes
1answer
3k views

Intersection of finite number of compact sets is compact?

Is the the intersection of a finite number of compact sets is compact? If not please give a counter example to demonstrate this is not true. I said that this is true because the intersection of ...
7
votes
1answer
938 views

Equivalence of reflexive and weakly compact

In a normed space $X$ is there an equivalence between these two proposition? $1)$ $X$ is reflexive; $2)$ $B$, the unit ball of $X$, is weakly compact.
5
votes
2answers
1k views

If the graph of a function $f: A \rightarrow \mathbb R$ is compact, is $f$ continuous where $A$ is a compact metric space?

I have seen answers to this question, which go beyond my understanding of compactness and continuity. I was wondering whether we can cook up a proof using sequential compactness and certain equivalent ...
1
vote
2answers
173 views

If $A$ is compact and $B$ is Lindelöf space , will be $A \cup B$ Lindelöf

I have 2 different questions: As we know a space Y is Lindelöf if each open covering contains a countable subcovering. (1) :If $A$ is compact and $B$ is Lindelöf space , will be $A \cup B$ ...
3
votes
1answer
76 views

Isomorphism isometries between finite subsets , implies isomorphism isometry between compact metric spaces

Let's $(X_1,d_1), (X_2,d_2)$ be compact metric spaces such that for every finite subset of $X_1$ like $A$ (respectively any finite subset of $X_2$ like $B$ ) there exists a finite subset of $X_2$ ...
8
votes
1answer
550 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 ...
6
votes
2answers
4k views

Prove: Every compact metric space is separable

How to prove that Every compact metric space is separable$?$ Thanks in advance!!
6
votes
2answers
807 views

Stone–Čech compactification of $\mathbb{N}, \mathbb{Q}$ and $\mathbb{R}$

I'm trying to find connections between Stone–Čech compactifications of $\mathbb{N}, \mathbb{Q}$ and $\mathbb{R}$, all with the euclidean topology. So, are there any ? e.g. is $\beta \mathbb{Q} = \beta ...
16
votes
2answers
913 views

Compactness of $\operatorname{Spec}(A)$

In an exercise in Atiyah-Macdonald it asks to prove that the prime spectrum $\operatorname{Spec}(A)$ of a commutative ring $A$ as a topological space $X$ (with the Zariski Topology) is compact. Now ...
8
votes
1answer
1k views

Metrizable compactifications

Suppose $X$ is a metric space. When does it have a metrizable compactification? Of course it is enough to discuss complete metric spaces, but separability may not be assumed here. I know that ...
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 ...
4
votes
6answers
2k views

Every compact metric space is complete

I need to prove that every compact metric space is complete. I think I need to use the following two facts: A set $K$ is compact if and only if every collection $\mathcal{F}$ of closed subsets with ...
3
votes
1answer
194 views

When is a subset of $\ell^2$ compact?

I have been looking on the internet for hours now and even asking in chat without an answer. When is a set $M\subseteq\ell^2$ compact? For $L^p$, there is the Arzelà–Ascoli theorem that provides a ...
1
vote
1answer
834 views

Compact spaces and closed sets (finite intersection property)

I am trying to prove the following theorem: A topological space $X$ is compact iff for every collection $\mathscr{C}$, of closed set in $X$ having the Finite Intersection Property (FIP), $\cap C$ of ...
1
vote
1answer
110 views

About $ \{ x \in[0,1]^{\omega_1}:|\{\alpha<\omega_{1} :x(\alpha)\ne 0 \}|\le\omega \}$

Take $X$ a Tychonoff product $[0,1]^{\omega_1}$ and as $Y$ the $\Sigma$-product $$ \{ x ∈[0,1]^{\omega_1}:|\{\alpha<\omega_{1} :x(\alpha)\ne 0 \}|\le\omega \}\;.$$ The space $X$ is compact by ...
32
votes
4answers
767 views

To show that the set point distant by 1 of a compact set has Lebesgue measure $0$

Could any one tell me how to solve this one? Let $K$ be a compact subset of $\mathbb{R}^n$, and $$A:=\{x\in\mathbb{R}^n:d(x,K)=1\}.$$ Show that $A$ has Lebesgue measure $0$. Thank you!
40
votes
6answers
2k views

Why is compactness in logic called compactness?

In logic, a semantics is said to be compact iff if every finite subset of a set of sentences has a model, then so to does the entire set. Most logic texts either don't explain the terminology, or ...
8
votes
5answers
779 views

Most astonishing applications of compactness theorem outside logic

The compactness theorem has a lot of applications to logic and model theory. I'm looking for applications. I'm looking for theorems in other areas of mathematics which seem at first sight to have ...
5
votes
1answer
174 views

Are there Hausdorff spaces which are not locally compact and in which all infinite compact sets have nonempty interior?

Here is the background material from which I am working: The Cantor set is an uncountable compact Hausdorff space with empty interior. In a locally compact Hausdorff space, each countable set has ...
5
votes
1answer
3k views

totally bounded, complete $\implies$ compact

Show that a totally bounded complete metric space $X$ is compact. I can use the fact that sequentially compact $\Leftrightarrow$ compact. Attempt: Complete $\implies$ every Cauchy sequence ...
4
votes
3answers
747 views

$[0,1]^{\mathbb{N}}$ with respect to the box topology is not compact

could anyone help to show that $[0,1]^{\mathbb{N}}$ with respect to the box topology is not compact? Thank you!
3
votes
2answers
348 views

Hausdorff space in which each point has a compact neighbourhood is locally compact

Could you help me prove the following fact? I've been trying to prove it and I've searched for a hint in Englking's book, but I haven't come up with anything: If $X$ is a Hausdorff space and each $x ...
15
votes
1answer
364 views

Cardinality of a locally compact Hausdorff space without isolated points

I am interested in the following result: Theorem. A locally compact Hausdorff topological space $X$ without isolated points has at least cardinality $\mathfrak{c}$. To prove it, one can find two ...
9
votes
3answers
726 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 ...
6
votes
2answers
181 views

Let $(M,d)$ be a compact metric space and $f:M \to M$ such that $d(f(x),f(y)) \ge d(x,y) , \forall x,y \in M$ , then $f$ is isometry?

Let $(M,d)$ be a compact metric space and $f:M \to M$ such that $d(f(x),f(y)) \ge d(x,y) , \forall x,y \in M$ ; then how to prove that $d(f(x),f(y))=d(x,y) , \forall x,y \in M$ i.e. that $f$ is an ...
3
votes
1answer
222 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 ...
7
votes
5answers
1k views

Topology: Example of a compact set but its closure not compact

Can anyone gives me an example of a compact subset such that its closure is not compact please? Thank you.
7
votes
3answers
729 views

$K\subseteq \mathbb{R}^n$ is a compact space iff every continuous function in $K$ is bounded.

I need to prove that $K\subseteq \mathbb{R}^n$ is a compact space iff every continuous function in $K$ is bounded. One direction is obvious because of Weierstrass theorem. How can i prove the other ...
4
votes
2answers
189 views

Compact space and Hausdorff space

A continuous map from a compact space to a Hausdorff space is closed. Why this is true? Help me please I want to learn why this is correct.
2
votes
2answers
2k views

$X$ compact metric space, $f:X\rightarrow\mathbb{R}$ continuous attains max/min

Let $X$ be a compact metric space, show that a continuous function $f:X\rightarrow\mathbb{R}$ attains a maximum and a minimum value on $X$. Attempt: So the important thing is that I have ...
-3
votes
6answers
846 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 ...
13
votes
5answers
9k views

Compact sets are closed?

I feel really ignorant in asking this question but I am really just don't understand how a compact set can be considered closed? I mean by definition of a compact set it means that given an open cover ...
10
votes
2answers
5k views

Understanding the definition of a compact set

I just need a bit of help clarifying the definition of a compact set. Let's start with the textbook definition: A set $S$ is called compact if, whenever it is covered by a collection of open sets ...