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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20
votes
5answers
9k 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 ...
40
votes
5answers
10k 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 ...
18
votes
5answers
4k 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 ...
46
votes
11answers
3k views

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}...
23
votes
2answers
3k 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 ...
11
votes
1answer
4k views

Prove the map has a fixed point

Assume $K$ is a compact metric space with metric $\rho$ and $A$ is a map from $K$ to $K$ such that $\rho (Ax,Ay) < \rho(x,y)$ for $x\neq y$. Prove A have a unique fixed point in $K$. The ...
10
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 ...
26
votes
12answers
9k views

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 $[a,b]$ ...
22
votes
1answer
3k 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 ...
103
votes
12answers
8k 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 ...
18
votes
2answers
6k 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 ...
16
votes
2answers
2k 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.
8
votes
2answers
3k 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* ...
13
votes
6answers
939 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 ...
5
votes
2answers
4k views

Cartesian product of compact sets is compact

Prove that if two sets $A$ and $B$ are compact then so is their Cartesian product $A \times B = \{(a,b): a \in A, b\in B\}$. The hint is to use Bolzano Weiertrass theorem and an argument of sequence ...
15
votes
3answers
2k 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
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 ...
16
votes
2answers
1k 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 ...
6
votes
6answers
5k 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 ...
2
votes
3answers
2k 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 ...
7
votes
1answer
2k 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.
7
votes
3answers
2k 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 ...
1
vote
1answer
2k 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 ...
55
votes
7answers
3k 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
2answers
323 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 ...
13
votes
2answers
2k 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 ...
7
votes
1answer
4k 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 ...
5
votes
1answer
5k views

Proving that $S=\{\frac{1}{n}:n\in\mathbb{Z}\}\cup\{0\}$ is compact using the open cover definition

Let $S=\{\frac{1}{n}:n\in\mathbb{Z}\}\cup\{0\}$ be a subset of $\mathbb{R}$. I have to prove using the open cover definition that this is compact. Could you help me, please?
1
vote
2answers
957 views

Proof: in $\mathbb{R}$, $((0,1),|\cdot|)$ is not compact.

Let $(M,d)$ be a metric space, and $A\subset M$. By definition, $A$ is said to be compact if every open cover of $A$ contains a finite subcover. What is wrong with saying that, in $\mathbb{R}$, if $I=...
5
votes
2answers
2k 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 ...
2
votes
1answer
430 views

When Cantor's Intersection theorem won't work with closed sets

Give an example to show that Cantor's Intersection Theorem would not be true if compact sets were replaced by closed sets. Compact set is closed and bounded, so what I'm going to find is something ...
2
votes
2answers
228 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 $A \cup B$ ...
1
vote
1answer
136 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 smaller ...
12
votes
6answers
2k 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.
11
votes
3answers
563 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 ...
10
votes
3answers
819 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 -...
10
votes
3answers
3k views

Inverse image of a compact set is compact

Let $X$ and $Y$ be topological spaces, $X$ compact, $f : X \to Y$ continuous. Then the preimage of each compact subset of $Y$ is compact. With the stipulation that $X$ and $Y$ are metric spaces, this ...
10
votes
2answers
755 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 ...
7
votes
2answers
1k 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 ...
8
votes
2answers
2k views

True Or not: Compact iff every continuous function is bounded [duplicate]

Let $X$ be a topological space. My question is: If $f:X\to \mathbb{R}$ is bounded for all such continuous $f$, then is $X$ compact. Is is really? If $X$ is the subset of $\mathbb{R}^d$, then it is ...
8
votes
2answers
2k 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 ...
7
votes
5answers
1k views

Fixed point Exercise on a compact set

Let $K$ a compact normed space and $f:K\rightarrow K$ so that $$\|f(x)-f(y)\|<\|x-y\|\quad\quad\forall\,\, x, y\in K, x\neq y.$$ Prove that $f$ have a fixed point.
0
votes
6answers
2k 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 ...
11
votes
2answers
2k views

Compact open sets which are not closed.

Can a nonclosed open subset of a $T_1$ topological space be compact? I mean an open compact set which is not clopen.
4
votes
1answer
277 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 ...
3
votes
4answers
5k views

Cover of (0,1) with no finite subcover & Open sets of compact function spaces

I just got back from my exam and these questions' solutions eluded me, it would be great to use the rest of my evening figuring these out... Q1: Find an open covering of the set $(0,1) \subset \...
3
votes
2answers
220 views

Show that $C_0([a, b], \mathbb{R})$ is not $\sigma$-compact

$C_0([a, b], \mathbb{R})$ is the space of real-valued continuous function on $[a, b]$. The hint says think Baire. So I assume that $C_0([a, b], \mathbb R)$ is $\sigma$-compact. Then it is the ...
-1
votes
3answers
226 views

Prove that $ S=\{0\}\cup\left(\bigcup_{n=0}^{\infty} \{\frac{1}{n}\}\right)$ is a compact set in $\mathbb{R}$.

Prove that $ S=\{0\}\cup\left(\bigcup_{n=0}^{\infty} \{\frac{1}{n}\}\right)$ is a compact set in $\mathbb{R}$, but $\bigcup_{n=0}^{\infty} \{\frac{1}{n}\}$ is not a compact set. (Can we use ...
3
votes
4answers
243 views

There's no continuous injection from the unit circle to $\mathbb R$

I read a proof that goes as follows: Let $U$ be the unit circle, and let $f : U \longrightarrow \mathbb R$ be a continuous mapping. $U$ is compact and connected, so $f(U)$ is a closed, bounded ...
2
votes
1answer
316 views

$G_\delta$ singletons in compact Hausdorff and first countability

Given a compact Hausdorff space $X$, if $a \in X$ is a $G_\delta$ singleton (i.e., $\{ a \}$ is a $G_\delta$ set), then there is a countable local base at $a$. The point $a$ can be written as ...