Tagged Questions

The compactness tag is for questions about compactness and its many variants (e.g. sequential compactness) as well locally compact spaces; compactifications (e.g. one-point, Stone-Cech) and other topics closely related to compactness.

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1answer
56 views

Compact metric space group $Iso(X,d)$ is also compact

Could you tell me how to prove that if metric space $(X,d)$ is compact, then the group $Iso(X,d)$ is also compact? The group $Iso(X,d)$ is considered with topology determined by a metric $\rho$ on ...
2
votes
1answer
35 views

Proof that the interior of any union of closed sets with empty interior in a compact Hausdorff space is empty

The question is pretty much in the title, I need to show that given $X$ is a compact Hausdorff space and $\left\{ A_n\right\}_{n=1}^\infty$ is a collection of closed subsets of $X$ each with empty ...
1
vote
1answer
28 views

The algebraic possibilities of the (topological) procedure of the compactification of a space

If $X$ is locally compact $K$-vector space, then $X\cup \{\infty\}$ is via the Alexandroff-compactification a compact space. But this purely topological procedure tells me nothing about the algebraic ...
4
votes
1answer
27 views

Characterization for compact sets in $\mathbb{R} $ with the topology generated by rays of the form $\left(-\infty,a\right) $

I'm trying to find a sufficient and necessary condition for a subset to be compact in $\mathbb{R} $ when the topology is generated by the basis $\left\{ \left(-\infty,a\right)\,|\, ...
-2
votes
3answers
51 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 ...
2
votes
1answer
30 views

Every countably compact, scattered $T_3$-space is sequentially compact

A space $X$ is called scattered provided every non-empty subspace $Y$ has an isolated point( with respect to the subspace topology on $Y$). How to prove that: Every countably compact, scattered ...
4
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0answers
53 views

Is $X$ pseudocompact

The following example with a little modified from the handbook of set theoretic topology, Page 574: Let $\kappa$ be any cardinal for which there exists a family $\{H_\alpha: \alpha < \kappa\}$ ...
0
votes
1answer
27 views

Can a Accumulation Point be an Eigenvalue?

I have a discrete (separable) infinite dimensional Hilbert Space with a compact operator defined on it. So 0 is an accumulation point (some theorem says so). Can 0 also be an eigenvalue? And how would ...
3
votes
1answer
35 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
0answers
67 views

How to show that $\exists~f\in C(X,\mathbb R)$ such that $f$ is unbounded? [duplicate]

Let $X$ be a non-compact metric space. How to show that $\exists~f\in C(X,\mathbb R)$ such that $f$ is unbounded?
0
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1answer
44 views

Unique nearest point in epsilon neighborhood of compact real manifold?

I have to proof the following assertion: Let $X$ be a compact submanifold of $\mathbb{R}^n$ and $\mathcal{U}^\varepsilon=\{p\in\mathbb{R}^n\;:\; |p-q|<\varepsilon \text{ for some }q\in X\}$. Then ...
2
votes
2answers
54 views

Is my proof correct? (minimal distance between compact sets)

I'm working out the following problem form Ahlfors' Complex Analysis text: "Let $X$ and $Y$ be compact sets in a complete metric space $(S,d)$. Prove that there exist $x \in X,y \in Y$ such that ...
1
vote
1answer
39 views

Returning to the starting point problem

I have a problem about a compact metric space $ \mathbf{X} $ in which there is a mapping from $ \mathbf{X}$ to $ \mathbf{x}$ and the distance function is such that $d( f(x), f(y)) = d(x, y)$. I ...
5
votes
1answer
65 views

Understanding the topological space $\beta \omega$

It is from the Handbook of Set-theoretic topology: We consider $\beta \omega$ as the set of all ultrafilters on $\omega$ with the topology which is generated by taking as a base all sets of the ...
3
votes
1answer
60 views

Compactness of the set of terms in a convergent (sub)net and its limit

In a Hausdorff topological space, let net $(x_d)_{d \in D}$ converge to $x$. The set $$ (\cup_{d \in D} \{x_d\}) \cup \{x\} $$ consisting of its terms and the limit need not be compact: in ...
15
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0answers
135 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 ...
5
votes
0answers
89 views

Is dependent choice necessary to prove every perfect compact Hausdorff space is uncountable?

The answer to Cardinality of a locally compact space without isolated point shows that AC is required to show that if $X$ is a compact Hausdorff space with no isolated points then $|X| \ge ...
3
votes
1answer
190 views

One point compactification of $[0,1] \times [0,1)$

Let $X = [0,1] \times [0,1) \subset \mathbb{R}^2$. I've already proven that this space is locally compact and found its one-point compactification but now I am stuck on the following; Let $Y = X \cup ...
-1
votes
1answer
143 views

Any net in $A\subseteq X$ has a cluster point in $X$. Is $\overline{A}$ compact?

Let $(X,\mathcal T)$ be a topological space, $A\subseteq X$ and any net in $A$ have a cluster point in $X$. Is $\overline{A}$ compact?
4
votes
2answers
96 views

How I can prove that: If $X$ is compact, then any map $f\colon X \to Y$ is proper?

A continuous map $f\colon X \to Y$ of locally compact spaces is called proper if for any compact $C\subset Y$ the preimage $f^{-1}(C)$ is compact. My question is: How I can prove that: If $X$ is ...
8
votes
1answer
101 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 ...
6
votes
3answers
139 views

Stone-Čech Compactification of the Natural numbers

I am trying to prove that if $U$ is contained in the Stone-Čech Compactification of the natural number ($\beta N$) that the closure of $U$ is open. I have a really hard time with even understanding ...
2
votes
1answer
49 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 ...
0
votes
1answer
55 views

A question on the compact subset of $[0,1]$

Let $S=\{K \subseteq [0,1]: K \text{ is compact and uncountable } \}$. How to see that $|S|=\mathfrak c$? Thanks for your help.
0
votes
0answers
39 views

Convex Hull of Precompact Subset is Precompact

I'm trying to prove that, if $K$ is a precompact (I've also heard the phrase totally bounded used for this) subset of a Banach Space $X$, then its convex hull is also precompact. I've come across a ...
1
vote
2answers
50 views

Inverse limit of compact metric spaces

Does an inverse limit of compact metric spaces need to be metrizable? When it is an inverse limit of a countable inverse system I know it is metrizable (even without compactness). But what if the ...
3
votes
1answer
51 views

How to show that metric space $(X,d)$ is complete

$(X,d)$ is a metric space with property that every closed and bounded set is compact.Now how can I show $X$ is complete? Can any one help me to give hints about it?
1
vote
2answers
44 views

Finding a bounded, non-compact set of functions $f:[0,1]\to\Bbb R $

Consider the metric space $(X, d)$ given by $$X = \{\text{all continuous functions}\,f:[0,1]\to\Bbb R\}$$ with $$d(f,g)=\sup_{t\in[0,1]}|f(t)-g(t)|.$$ Find with proof a set $A \subseteq X$ with ...
4
votes
4answers
67 views

How can I prove that if $A$ is compact, then $A$ if finite? (Under the discrete metric)

Let $\delta$ be the discrete metric on a non-empty set $X$. Characterize the subsets of $X$ which are compact in $(X, \delta)$. I remember the answer from a previous class: $A \subseteq X$ is ...
14
votes
2answers
192 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, ...
3
votes
1answer
70 views

Tychonoff Theorem and the axiom of choice

How to show that The Tychonoff Theorem and the axiom of choice are equivalent? Here I want to collect ways to prove it. Thanks for your help.
1
vote
1answer
63 views

If $X$ is complete and totally bounded, then $X$ is compact [closed]

Let $X$ be a metric space. Whar is your favorite way to show: If $X$ is complete and totally bounded, then $X$ is compact? Thanks for your help.
3
votes
2answers
52 views

A $T_2$ space is locally compact iff..

A $T_2$ space is locally compact iff it has a base $\beta $ s.t $\forall B\in \beta $ we have $\bar B$ is compact. My definition of locally compact is that $\forall x\in X$ has a compact neighborhood ...
16
votes
5answers
290 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 ...
6
votes
2answers
64 views

Question on compactification

I was studying for quals and had trouble with this question. Any help would be great, thanks. A two-point compactifcation of a Hausdorff space $X$ is a compact Hausdorff space $Y$ such that $X$ is a ...
4
votes
0answers
190 views

If $f: X \to Y$, when do we have $\beta Y \supset \overline{f(X)} = \beta X$?

Suppose that $X$ and $Y$ are Tychonoff spaces, denote by $\beta X$ and $\beta Y$ their Stone-Čech compactifications and let $f:X\to Y$ be a continuous map. Using the embedding $Y\hookrightarrow\beta ...
8
votes
1answer
106 views

Clopen subsets of a compact metric space

I am aked to show that in a compact metric space we can find at most countably many subsets which are both: open and close. I would be grateful for your help.
3
votes
3answers
134 views

$f:X\to X$ is one-one and continuous on a compact space. Is $f$ surjective?

Let $(X,\mathcal T)$ be a compact Hausdorff topological space and $f:X\to X$ be one-to-one and continuous. Is $f$ surjective?
2
votes
2answers
32 views

Question about finite sets/compactness

I understand that every finite subset of a metric space is compact. But are there any topological spaces where finite sets are not compact? Is that even possible? I don't think it is but I just want ...
2
votes
1answer
43 views

On compact topological group

Must a compact topological group be metrizable? If not, is it separable? Thanks for any help.
13
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10answers
1k 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 $[a,b]$ ...
3
votes
2answers
32 views

Is every countably compact space feebly compact?

A topological space is said to be feebly compact if every locally finite cover by nonempty open sets is finite. Every compact space is feebly compact but how about countably compact spaces?
5
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1answer
158 views

Under what condition only does every compact subset of $X$ is closed implies $X$ Hausdorff?

It is trivial to see that: If $X$ is Hausdorff, then every compact subset of $X$ is closed. I am asking under what condition does the converse hold, i.e. when does If every compact subset of $X$ is ...
1
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2answers
65 views

Is it true that a metric compact is complete space?

Is it true that a metric compact is complete space? I think that it is true.
1
vote
2answers
50 views

Hausdorff space and Cantor's intersection theorem

$X$ is a Hausdorff space, $C_i$ is a non-empty closed subset of $X$ and $C_{k+1}\subseteq C_k$ , show that $\displaystyle \bigcap_{i\in \mathbb{N}} C_i$ is compact. I tried to prove by ...
6
votes
0answers
70 views

An example of a compact multiplicatively unbounded ring

My teacher asked me to build an associative topological Hausdorff compact ring $R$ with $1$, which is multiplicatively unbounded. That means there is a neighborhood $U\ni 1$ such that $FU\not=R$ for ...
4
votes
2answers
57 views

one point compactification

I am asked to describe the one point compactification of $(0,1) \cup [2,3)$ of $\Bbb R$ and if I'm not mistaken it is just a circle union the closed set [2,3] correct? Am I missing something?
3
votes
0answers
41 views

Another question in relation to Tychonoff theorem

Let $X_i$ be compact topological spaces and let $X = \prod_{i \in I}X_i$ and let $\mathscr F$ be ultrafilter on $X$. Define $\mathscr F_i = \{Y \subseteq X_i : \pi_i^{-1}Y \in \mathscr F\}$. Here ...
2
votes
1answer
91 views

How to prove a topologic space $X$ induced by a metric is compact if and only if it's sequentially compact?

A topological space $X$ is called sequentially compact if every sequence of points in $X$ has a subsequence that converges to a point in $X$. I know it's very similar to Bolzano–Weierstrass theorem ...
4
votes
2answers
91 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.

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