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-Cech) and other topics closely related to compactness.

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63
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12answers
5k 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 ...
37
votes
6answers
1k 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 ...
28
votes
4answers
588 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!
25
votes
1answer
401 views

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 ...
24
votes
5answers
4k 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 ...
22
votes
6answers
832 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
2answers
236 views

Paracompactness of CW complexes (rather long)

I finished reading Lee's 'introduction to topological manifolds' (2nd edition) and I'm currently tying up some loose ends. One thing I can't understand is the proof of paracompactness of CW complexes. ...
19
votes
1answer
206 views

If $S\times\mathbb{R}$ is homeomorphic to $T\times\mathbb{R}$, and $S$ and $T$ are compact, can we conclude that $S$ and $T$ are homeomorphic?

If $S \times \mathbb{R}$ is homeomorphic to $T \times \mathbb{R}$ and $S$ and $T$ are compact, connected manifolds (according to an earlier question if one of them is compact the other one needs to be ...
16
votes
11answers
3k 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 $[a,b]$ ...
15
votes
1answer
322 views

Can compacts on a real line behave “paradoxically” with respect to unions, intersections, and translation? What about other topological groups?

I have the following question i cannot answer myself. Consider two compacts $A$ and $B$ on the real line $\mathbb R$. Let $A'$ be a translation of $A$ and $B'$ a translation of $B$: $A' = A + a$, ...
14
votes
8answers
2k views

How to understand compactness? [duplicate]

How to understand the compactness in topology space in intuitive way?
14
votes
2answers
281 views

Is there a “tree-like” proof of compactness theorem in the case of uncountably many variables?

I like proofs using trees and König's lemma, since they are very visual. One of the applications of König's lemma you can show to students is proving compactness theorem for propositional calculus, ...
14
votes
1answer
299 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 ...
13
votes
2answers
273 views

Given a fiber bundle $F\to E\overset{\pi}{\to} B$ such that $F,B$ are compact, is $E$ necessarily compact?

Consider a (locally trivial) fiber bundle $F\to E\overset{\pi}{\to} B$, where $F$ is the fiber, $E$ the total space and $B$ the base space. If $F$ and $B$ are compact, must $E$ be compact? This ...
12
votes
5answers
355 views

Compactness in $\mathbb{Q}$

Proving that $[0,1]\subset\mathbb{R}$ is compact you make use of completeness and this is a fundamental step in order to characterize compact subsets of $\mathbb{R}$. Trying to state an analogous ...
12
votes
2answers
2k views

Topology of matrices

1.Consider the set of all $n×n$ matrices with real entries as the space $\mathbb R^{n^2}$ . Which of the following sets are compact? (a) The set of all orthogonal matrices. (b) The set of all ...
12
votes
3answers
765 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 ...
11
votes
1answer
174 views

Does there exist a topology for a set $X$ which is compact and Hausdorff?

For every set $X$ and every topology $\tau$ over $X$ we have that $\tau$ contains the trivial topology $\{ X, \emptyset\}$, which is compact, and is contained in the discrete topology $\{ S: S ...
11
votes
2answers
174 views

The smallest compactification

I've been reading about compactifications, and I am very familiar with the Stone-Čech compactification, which can be thought of as the largest compactification of a space. For a locally compact space ...
11
votes
2answers
159 views

Are compact spaces characterized by “closed maps to Hausdorff spaces”?

It is well known that any continuous map between from a compact space to a Hausdorff space must be a closed map. Does this fact characterize compactness? That is, if for a space $X$, every continuous ...
11
votes
5answers
2k 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 ...
11
votes
1answer
122 views

Proving that a metric space is a group

I'm stuck on this relatively hard problem. Let $G$ be a non-empty set, $d$ a distance on $G$ and $\cdot$ an associative operation on $G$ $\cdot$ is such that $$\forall a \in G , \forall x \in G ...
10
votes
2answers
177 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 ...
10
votes
1answer
181 views

Growth $\beta X\setminus X$ of a Banach space $X$

Is there an analytic characterisation of the Čech-Stone compactification (in the norm topology, which is a normal space) of a Banach space $X$? The reason I ask is because I want to know what the ...
10
votes
1answer
1k views

If $A$ and $B$ are compact, then so is $A+B$.

This is an exercise in Chapter 1 from Rudin's Functional Analysis. Prove the following: Let $X$ be a topological vector space. If $A$ and $B$ are compact subsets of $X$, so is $A+B$. My guess: ...
9
votes
4answers
3k 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 ...
9
votes
6answers
457 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 ...
9
votes
2answers
3k 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 ...
9
votes
2answers
277 views

Can compact sets completey determine a topology?

Suppose that $\tau_1$ and $\tau_2$ are two topologies on a set $X$ with the property that $K\subset X$ is compact with respect to $\tau_1$ if and only if $K$ is compact with respect to $\tau_2$. Then ...
9
votes
1answer
320 views

How many points does Stone-Čech compactification add?

I would like to know how Stone-Čech compactification works with simple examples, like $(0,1)$, $\mathbb{R}$, and $B_r(0)$ (the open ball of $R^2)$. I've studied the one-point compactification and this ...
9
votes
3answers
655 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 ...
9
votes
2answers
385 views

If $A$ is compact, is then $f(A)$ compact?

I just got my exam back, and I still cannot understand this question: Given a continuous function $f:A\subseteq\mathbb{R}\to\mathbb{R}$, show that if $A$ is a compact set, then its image, $f(A)$, ...
9
votes
1answer
352 views

$(X,\mathscr T)$ is compact $\iff$ every infinite subset of $X$ has a complete limit point in $X$.

Let $(X,\mathscr T)$ be a topological space. Given $A\subseteq X$, we say that $x$ is a complete limit point of $A$ if for every nbhd $N$ of $x$, $|N\cap A|=|A|$. I want to prove Suppose ...
9
votes
1answer
192 views

Are there non-Hausdorff examples of maximal compact topologies in the lattice of topologies on a set?

In the lattice of topologies on a set $X$, the compact topologies are a lower set in the lattice, while the Hausdorff topologies are an upper set. A result of this theorem is that the compact ...
9
votes
3answers
281 views

A question on a compact space

Show: If the closure of every discrete subset of a space is compact then the whole space is compact. Thanks advance:)
9
votes
1answer
178 views

Ideal in compact Hausdorff space

This is exercise 70, chapter 4. from Folland (page 142) Let $X$ be a compact Hausdorff space. An ideal in $C(X, \mathbb{R})$ is a subalgebra $J$ of $C(X, \mathbb{R})$ such that if $f\in J$ and $g\in ...
9
votes
1answer
390 views

Prove that a compact metric space can be covered by open balls that don't overlap too much.

The problem is: For compact metric space $(X,d)$ prove that for every $r>0$ there exists a subset $S$ of $X$ such that $\{\mbox{Open balls of radius }r\mbox{ centered at }p \mid\mbox{ for all }p ...
8
votes
7answers
593 views

Give an example of a simply ordered set without the least upper bound property.

In Theorem 27.1 in Topology by Munkres, he states "Let $X$ be a simply ordered set having the least upper bound property. In the order topology, each closed interval in $X$ is compact." (The LUB ...
8
votes
5answers
751 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 ...
8
votes
1answer
429 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 ...
8
votes
3answers
285 views

A theorem due to Gelfand and Kolmogorov

For any topological space $X$, we can define $C(X)$ to be the commutative ring of continuous functions $f\,:\,X\rightarrow \mathbb{R}$ under pointwise addition and multiplication. Then $C(-)$ becomes ...
8
votes
2answers
222 views

Compact maps problem in Lax

In Functional Analysis of Peter Lax there are the following exercise Show that if $\bf C$ is compact and $\{{\bf M}_n \}$ tends strongly to $\bf M$, then $\bf CM_n$ tends uniformly to $\bf CM$. ...
8
votes
2answers
669 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 ...
8
votes
3answers
113 views

Is this kind of space metrizable?

It has a nice result from Tkachuk V V. Spaces that are projective with respect to classes of mappings[J]. Trans. Moscow Math. Soc, 1988, 50: 139-156. If the closure of every discrete subset of a ...
8
votes
1answer
2k views

Rationals are not locally compact and compactness

I was wondering if someone can please help me with the following problems: Show that $\mathbb{Q}$ is not locally compact. Prove that if $X$ is Lindelöf and $Y$ is compact then $X \times Y$ is ...
8
votes
1answer
285 views

Alexandroff compactification question

If $X$ is a locally compact and metrizable space such that its Alexandroff compactification is not first countable. Does this imply that no other compactification of $X$ can be first countable? Why?
8
votes
1answer
151 views

One question related with sequential compactness

While trying to prove that a set is sequentially compact, I was suggested to prove by contradiction -- this is how it went, at least part of it: Definition. We say that a set $A$ is sequentially ...
7
votes
5answers
4k 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 ...
7
votes
5answers
388 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 ...
7
votes
2answers
107 views

dense subspace of $\beta \Bbb N \times \beta \Bbb N$

Let $\beta \Bbb N$ be a Čech-Stone compactification of the discrete space $\Bbb N$ and fix a point $p\in \beta \Bbb N\setminus \Bbb N$. Put $X=\Bbb N\cup \{p\}$ and $Y=\beta \Bbb N \setminus \{p\}$. I ...