Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [compactness]

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. This includes logical compactness.

Compactness is a topological property. We say that a topological space $X$ is compact if whenever we cover $X$ by a collection of open sets we can find a finite number of open sets from the collection which cover $X$. For example, $[0,1]$ is a compact subspace of $\mathbb{R}$, but $(0,1)$ and $\mathbb{R}$ are not.

We say that a space $X$ is sequentially compact if every sequence has a convergent subsequence. These properties are equivalent for metric space, although neither implies the other in general.

This tag may also be used for questions about logical compactness, such as the compactness theorem.

More information can be found on Wikipedia.

5759 questions
202
votes
13 answers

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 theorem, and a proof continuous functions on R from…
FireGarden
  • 5,489
  • 4
  • 22
  • 33
132
votes
14 answers

What should be the intuition when working with compactness?

I have a question that may be regarded by 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 $\mathbb{R}^n$ the compact sets are those that are closed…
Gold
  • 24,123
  • 13
  • 78
  • 179
93
votes
2 answers

A and B disjoint, A compact, and B closed implies there is positive distance between both sets

Claim: Let $X$ be a metric space. If $A,B\subset X$ are disjoint, $A$ is compact, and $B$ is closed, then there is $\delta>0$ so that $ |\alpha-\beta|\geq\delta\;\;\;\forall\alpha\in A,\beta\in B$. Proof. Assume the contrary. Let $\alpha_n\in…
Benji
  • 5,176
  • 5
  • 24
  • 27
88
votes
4 answers

Projection map being a closed map

Let $\pi: X \times Y \to X$ be the 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 necessary or do we have any other weaker…
M.Subramani
  • 1,231
  • 1
  • 12
  • 13
85
votes
7 answers

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 allude to the topological property of compactness. I…
vanden
  • 1,157
  • 9
  • 11
79
votes
5 answers

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 googling and wikipedia checks will turn up examples…
matt
  • 1,925
  • 1
  • 15
  • 15
73
votes
5 answers

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. By definition of a compact set it means that given an open cover we can find a finite subcover the covers the…
InsigMath
  • 1,895
  • 2
  • 17
  • 25
64
votes
12 answers

How to prove every closed interval in R 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 $[a,b]$ is compact. Thanks for your help and any link.
Paul
  • 19,906
  • 5
  • 33
  • 70
54
votes
3 answers

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 finite number of compact sets are closed. Which…
John Buchta
  • 551
  • 1
  • 5
  • 5
53
votes
3 answers

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 contrapositive. Realistically, this may be the…
combinator
  • 1,336
  • 1
  • 11
  • 16
45
votes
2 answers

How to prove that a compact set in a Hausdorff topological space is closed?

How to prove that a compact set $K$ in a Hausdorff topological space $\mathbb{X}$ is closed? I seek a proof that is as self contained as possible. Thank you.
Elias Costa
  • 13,301
  • 4
  • 43
  • 79
41
votes
3 answers

Difference between closed, bounded and compact sets

In real analysis, there is a theorem that a bounded sequence has a convergent subsequence. Also, the limit lies in the same set as the elements of the sequence, if the set is closed. Then when metric spaces are introduced, there is a similar…
user125395
  • 525
  • 1
  • 6
  • 4
38
votes
4 answers

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!
Marso
  • 31,275
  • 18
  • 107
  • 243
36
votes
1 answer

Show that a proper continuous map from $X$ to locally compact $Y$ is closed

Let $f: X \to Y$ be continuous and proper (a map is proper iff the preimage of a compact set is compact). Furthermore, assume that $Y$ is locally compact and Hausdorff (there are various ways of defining local compactness in Hausdorff spaces, but…
JZS
  • 4,468
  • 1
  • 16
  • 39
36
votes
1 answer

When is Stone-Čech compactification the same as one-point compactification?

For the space $\omega_1$ (with the order topology) we have $\beta\omega_1=\omega_1+1$ (or $\beta[0,\omega_1)=[0,\omega_1]$, if you prefer this notation), i.e., it is an example of a space for which Stone-Čech compactification and one-point…
Martin Sleziak
  • 50,316
  • 18
  • 169
  • 342
1
2 3
…
99 100