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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9
votes
4answers
4k 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
833 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 ...
27
votes
5answers
5k 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 ...
14
votes
8answers
2k views

How to understand compactness? [duplicate]

How to understand the compactness in topology space in intuitive way?
9
votes
2answers
772 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 ...
22
votes
6answers
950 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 ...
12
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 ...
12
votes
2answers
394 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 ...
5
votes
2answers
862 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 ...
19
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 ...
5
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* ...
9
votes
1answer
2k 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 ...
8
votes
1answer
455 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 ...
10
votes
6answers
527 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 ...
3
votes
1answer
147 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 ...
2
votes
3answers
709 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 ...
1
vote
1answer
49 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 ...
1
vote
1answer
99 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 ...
64
votes
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 ...
5
votes
1answer
133 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 ...
14
votes
1answer
314 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
664 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 ...
3
votes
1answer
2k 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 ...
6
votes
2answers
673 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 ...
4
votes
2answers
154 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.
3
votes
2answers
207 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 ...
9
votes
2answers
4k 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
1answer
3k 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
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 ...
6
votes
3answers
441 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 ...
6
votes
1answer
786 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 ...
5
votes
1answer
617 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.
4
votes
3answers
324 views

Axiom of choice and compactness.

I was answering a question recently that dealt with compactness in general topological spaces, and how compactness fails to be equivalent with sequential compactness unlike in metric spaces. The only ...
2
votes
1answer
184 views

Dose pointwise equicontinuous and uniformly equicontinuous implies compactness?

If every sequence of pointwise equicontinuous functions $M \rightarrow \mathbb{R}$ is uniformly equicontinuous, dose this imply that $M$ is compact ?
-1
votes
3answers
120 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 ...
4
votes
2answers
98 views

Contraction of compact sets

I am trying to solve the following problem. Let $X$ be a compact Hausdorff space and let $f:X\to X$ be continuous. Show that there exists a non-empty set $A\subset X$ such that $f(A)=A$. ...
4
votes
1answer
71 views

Is a minimal Hausdorff uniformity compact?

Let $(X,\mathcal D)$ be a Hausdorff uniform space and for each Hausdorff uniformity $\mathcal U$ on $X$, $$\mathcal U \subseteq\mathcal D\to \mathcal U =\mathcal D$$ Is $(X,\mathcal D)$ compact?
3
votes
2answers
71 views

Direct proof of compactness of $\mathbb{Z}_p$

Let $\mathbb{Z}_{p}$ be completion of $\mathbb{Z}$ with respect to $p-$norms. Actually I know that $\mathbb{Z}_{p}$ is bijective to Cantor set, which is compact, therefore by homeomorphism, it is also ...
3
votes
1answer
505 views

For two disjoint compact subsets $A$ and $B$ of a metric space $(X,d)$ show that $d(A,B)>0.$ [duplicate]

I was thinking about the following problem: For two disjoint compact subsets $A$ and $B$ of a metric space $(X,d)$ show that $d(A,B)>0.$ I'm having doubt with my attemp. Please have a look and ...
3
votes
1answer
271 views

Are all compact sets in $ \Bbb R^n$, $G_\delta$ sets?

Are all compact sets in $\Bbb R^n$, $G_\delta$ sets? I know that compact set is bounded and closed.
3
votes
2answers
3k views

Prove: Every compact metric space is separable

How to prove that Every compact metric space is separable$?$ Thanks in advance!!
2
votes
1answer
49 views

infinite disceret subspace

**Each infinite subspace of a KC space contain an infinite discrete subspace.** Proof: Let $ (X,\tau)$ be aKC space, and $A ‎‎\subseteq‎ X$ is infinite. since $A$ does not have the cofine topolog , ...
2
votes
2answers
148 views

I need to show that $K$ is compact and that $co(K)$ is bounded, but not closed.

Let $x_n$ be a sequence in a Hilbert space such that $\left\Vert x_n \right\Vert=1$ and $ \langle x_n,\ x_m \rangle =0 $, for all $n \neq m$. Let $ K= \{ x_n/ n : n \in \mathbb{N} \} \cup \{0\} $. ...
1
vote
1answer
159 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
28 views

A hereditarily Lindelöf $KC$-space $( X,τ )$ is Katětov-$KC$ if and only if there is a weaker sequential $US$ topology $σ⊂τ

A space $( X,τ )$ is said to be Katětov $ KC $ if there is a topology $ σ⊂τ$ such that $( X,σ )$ is minimal $ KC $. The notion of strongly KC-spaces, that is, those spaces in which every ...
1
vote
1answer
140 views

the diameter of nested compact sequence

Let $E_{i}$ be a nested compact subsequence s.t $\forall i E_i\geqslant r$ for $r>0$. How can we show that this implies that $\bigcap_{i=1}^\infty E_i$ also has parameter bigger than $r$?
1
vote
2answers
252 views

Heine-Borel Theorem ($\mathbb{R}^k$) (in ZF)

Heine-Borel Theorem; If $E \subset \mathbb{R}^k$, then $E$ is compact iff $E$ is closed and bounded. I have proved 'closed and bounded⇒compact' and 'compact⇒bounded'. (There exists $r\in \mathbb{R}$ ...
1
vote
2answers
1k 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 ...
0
votes
2answers
82 views

$ KC $ spaces imply $ US $ spaces , but vise versa is false.

In the $ US $ space , each convergent sequence has unique limit. In the $ KC $ space , every compact subset is closed. It easy to show that $ KC $ spaces imply $ US $ spaces. The ...