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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1answer
16 views

Show that, if $A\subset X\subset\mathcal{l}^{2}$, then $X$ is not pre-compact on $(\mathcal{l}^{2},\lVert \rVert)$.

For each $n\in\mathbb{N}$ let $e_{n}=(x_{k})_{k\in\mathbb{N}}$ with $x_{n}=1$ and $x_{i}=0$ for all $i\neq n$. Then $A=\{e_{n}:n\in\mathbb{N}\}\subset\mathcal{l}^{2}$ and $d(e_{n},e_{m})=\lVert ...
3
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1answer
54 views

If $X$ is a metric, then $X$ is compact if and only if $X$ is sequentially compact - axiom of choice usage

I'm going through a proof for the theorem: If $X$ is a metric, then X is compact if and only if X is sequentially compact. I have already posted this here. However this time I'm looking at the ...
1
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1answer
39 views

If $X$ is a metric, then $X$ is compact if and only if $X$ is sequentially compact

I'm going through a proof for the theorem: If $X$ is a metric, then $X$ is compact if and only if $X$ is sequentially compact. I'm trying to understand the easier forward direction but I'm ...
0
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2answers
49 views

Exercise about closed and compact sets from metric space

I have this exercise; First part: Let $E$ be a metric space, and $(F_n)$ a decreasing sequence of closed set from $E$ and let $(x_n)$ a convergent sequence such that $x_n\in F_n, $for all $n\geq0$. ...
3
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0answers
38 views

Infinite Cartesian Product, Metric Triangle Inequality

For $\{X_j : j\in \mathbb{Z}^+\},$ each compact metric spaces, the infinite Cartesian product metric space is defined as $$X = \prod_{j=1}^{\infty} X_j$$ We make X a metric space by setting $$d(x,y) ...
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0answers
50 views

Convex and continuous function on compact set implies Lipschitz

Let the function $f: C \rightarrow \mathbb{R}$ be convex and continuous, where $C \subset \mathbb{R}^n$ is a compact set. Prove or disprove that $f$ is Lipschitz continuous on $C$. Comments: If $f$ ...
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2answers
42 views

Generalization of cantors intersection theorem

Let $A_1\supset A_2\supset\cdots$ be a sequence of connected compact subsets of $\mathbb{R}^2$. Is it true that their intersection $A=\bigcap_{i=1}^{\infty}A_i$ is connected also? Suppose it is not ...
0
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1answer
24 views

A result using compactness and LUB axiom

Let $a,b \in \mathbb{R}, a<b$ and $\mathcal{A}$ be a collection of open sets in $\mathbb{R}$ such that $[a,b] \subseteq \cup_{A \in \mathcal{A} }A, C=\{x \in [a,b] : [a,x]$ is covered by finitely ...
1
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1answer
39 views

For $f \in C(X)$, if $\alpha(f+c)$ belongs to $\overline{\mathcal{A}}$, then $f$ also belongs to $\overline{\mathcal{A}}$

Let $\mathcal{A}$ be an algebra of continuous real-valued functions on a compact space $X$ that contains the constant functions. Let $f \in C(X)$ have the property that for some constant function $c$ ...
1
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1answer
22 views

General Triangle Inequality of a function

Let $\phi : [0,\infty) \rightarrow [0,\infty)$ have the following properties: Assume $$\phi(0)=0, \phi(s)<\phi(s+t)\leq \phi(s)+\phi(t)$$ with $ s\geq0,t>0$. Prove that if $d(x,y)$ is ...
2
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1answer
50 views

On the Banach–Alaoglu theorem: is the unit ball of an equivalent norm also weak-* compact?

Suppose that $E$ is a Banach space and let $E^*$ denote its dual space with canonical norm $\lVert\bullet\rVert_{E^*}$. Suppose that $\lvert\bullet\rvert_{E^*}$ is an equivalent norm on $E^*$. The ...
0
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1answer
22 views

Show that there is point such that the ray connecting it to the origin has the maximum slope

Let $S$ be a compact subset of the open first quadrant of the plane. Show that there is point $p_0=(x_0,y_0)$ in $S$ such that the ray connecting it to the origin has the maximum slope. Is this true ...
1
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1answer
40 views

Finite union of compact sets

Let $K_1,K_2,\ldots,K_N$ be compact subsets of the metric space $(X,d)$. Now I need to show that: $K_1\cup K_2 \cup \cdots \cup K_N$ is compact. My Attempt: I have the definiton: Let $(X,d)$ be ...
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0answers
40 views

Is this a compact manifold?

Consider $X=\left\{0,1,2\right\}^{\mathbb{Z}}$. My very short, and hopefully not too stupid question is, if $X$ is a compact manifold. I think compact is clear by Tychonoff's theorem, but I do not ...
0
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1answer
34 views

How can I prove this version of tube lemma for Tychonoff theorem?

I am trying to prove this version of tube lemma for the Tychonoff theorem. Lemma. Let $\mathscr{A}$ be a collection of basis elements for the topology of the product space $X \times Y$ , such that no ...
1
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1answer
27 views

Does separately continuous on compact set imply boundedness?

Let $f: \overline{\Omega} \times I \rightarrow \mathbb{R}$, is a continuous function on both variables (separately continuous). $\Omega \in \mathbb{R}^n$ is open, bounded. $I$ is a closed interval in ...
2
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1answer
27 views

Where the dense property applied in this proof using compactness?

There exists $t_1, t_2, \ldots, t_k \in \mathbb{Q}\cap[a,b]$ such that $\forall x \in [a,b],\ |x-t_j|<\delta$, for at least one $j = 1,\ldots, k$ My professor provide me with the hint that ...
0
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2answers
65 views

Maps on the hyperspace of compact sets

In the theory of fractals via iterated function systems, it is well-known that an IFS $\{f_i\}_{i=1}^n$ (being a finite collection of contractions defined on a metric space $X$) induces a single map ...
4
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0answers
61 views

Is this property of continuous maps equivalent to properness?

For the purposes of my question, a continuous map $f : X \to Y$ is proper if it is closed and the preimage of every compact subspace of $Y$ is a compact subspace of $X$. Say a continuous map $f : X ...
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3answers
37 views

prove: difference of compact set and open set is compact

claim: Let $(M, d)$ be a metric space and $K \subset M$ compact, $O \subset M$ open. Show that $K - O$ is compact. Proof: I think this should follow directly. If $K$ is a compact set, that means ...
0
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1answer
34 views

Image of a precompact under the action of continuous function is a precompact

Suppose $(X, d_x)$ and $(Y, d_y)$ be two metric spaces and $f\colon X \to Y$ be a continuous function . The problem is to prove that image $f(A)$ of every precompact $A \subset X$ is also a ...
2
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1answer
44 views

The space of continuous fuctions is compact - Other direction!

we all know that if $X$ is a compact topological space then $F(X)$ is compact for all $F\colon X\to \mathbb{R}$ continuous. I was wondering whether the converse is true? For metric spaces I have found ...
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1answer
45 views

Strangely defined ball compact in $L^p(I)$ or not?

Let $I = (0, 1)$ and $1 \le p \le \infty$. Set$$B_p = \{u \in W^{1, p}(I) : \|u\|_{L^p(I)} + \|u'\|_{L^p(I)} \le 1\}.$$When $1 < p \le \infty$, does it necessarily follow that $B_p$ is compact in ...
1
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1answer
45 views

One-point compactification of a locally connected space.

Is the one-point compactification of a connected and locally connected space also locally connected? My guess is no, because I haven't been able to prove it. But of course I also haven't come up with ...
8
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2answers
140 views

Points in the boundary of a compact set $K\subset\mathbb{R}^2$ reachable by a path in $K^c$

Let $K\subset\mathbb{R}^2$ be compact. Let the path boundary of $K$ denote the set of points in $z\in K$ such that for some point $w\in K^c$, there is a continuous path $\gamma:[0,1]\to\mathbb{R}^2$ ...
0
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2answers
22 views

Proving non-compactness of a manifold

I have been trying to solve the following problem: Let $M \subset \mathbb R^3$ be the set of points $(x,y,z) \in \mathbb R^3$ at which $xy + xz + yz = 1.$ Prove that $M$ is a $2$-dimensional manifold. ...
3
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1answer
30 views

a statement in the signature $\{c,f^1, R^2\}$ is satisfiable in a structure M $\iff $ for every element there exists a closed term t s.t. $t^M=a$

Question: Prove/Disprove that there exists a statement A in the signature $\{c,f^1, R^2\}$ that is satisfiable in a structure M $\iff $ for every element in $a\in D^M$ there is a closed term t s.t. ...
3
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1answer
46 views

Prove that there doesn't exist a statement in first order logic which is valid iff G is a 3-sparse graph

Question: A graph G is a 3-sparse graph if in every finite subgraph of $G$, the number of edges is at most 3 times the number of vertices. Prove that there doesn't exist a statement over the ...
0
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1answer
32 views

Example of an open cover of $(0, 1)$ with no finite subcover

Question: Let $F$ be the interval $(0,1)$ and find an open cover $G$ such that no finite sub-collection of $G$ covers $F$. I believe I have the answer I would appreciate some reassurance of my answer ...
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4answers
216 views

Compactness of $Y$ implies compactness of $X$

Question is as follows : Suppose $p:X\rightarrow Y$ is a closed continuous surjection such that $p^{-1}(\{y\})$ is compact for each $y\in Y$. Show that if $Y$ is compact then $X$ is compact. Hint : ...
3
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3answers
48 views

Lebesgue measure of a set

Question is : Let $K\subseteq \mathbb{R}^n$ is compact then $M=\{x\in \mathbb{R}^n : d(x,K)=1\}$ is of measure zero.. I did the following for $n=1$.. As $K$ is closed, $\mathbb{R}\setminus K$ is ...
0
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2answers
67 views

Is the ball compact?

Consider the space $C[0,1]$ of the continuous functions $f\colon [0,1]\to \Bbb R$, with $d_\infty(f,g)= \max_{x\in [0,1]} \lvert f(x)-g(x) \rvert$. Is the unit ball $\bar B _1 (0)$ compact, where ...
3
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3answers
70 views

Is there a countably compact sequential non-$T_2$ space that is not sequentially compact?

Let $X$ be a topological space. Definitions: $X$ is countably compact if every countable open cover of $X$ has a finite subcover or equivalently, every sequence in $X$ has a cluster point. $X$ is ...
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4answers
73 views

Infimum of distance between two sets

I have come across one little snippet in my Real Analysis book and I can not get it through. It generally says: $E\subset [0,1] \subset G$ and $E$ is closed $G$ is open. Define ...
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0answers
31 views

Is the collection of compact subsets of an open and convex set non-empty?

Consider an open and convex set $B \subseteq \mathbb{R}^k$. Is the collection of compact subsets of $B$ non-empty?
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0answers
37 views

Every compact subset of an open convex set $B\subset\mathbb{R}^k$ can be covered by finitely many cubes

Consider the following statement: Every compact subset of an open convex set $B\subset\mathbb{R}^k$ can be covered by finitely many cubes having edges parallel to the coordinate directions ...
2
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2answers
36 views

Give an example of a equicontinuous that does not converge uniformly

Give an example of a equicontinuous sequence of functions ($f_n$) over a non-compact set $S\subset\Bbb R^n$ converging pointwise to a function $f$ at each $x\in S$, but $f_n$ does not converge ...
0
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1answer
56 views

Proving that a set is not compact directly from the definition

Prove that the disk $D(a;R)=\{z:\lvert z-a \rvert<R \} $ is not compact. I know that we can prove that the set is not closed or not bounded and we can deduce directly that the set is not compact ...
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2answers
72 views

Logical trap in R topology

We know that in metric spaces, Bolzano-Weierstrass (BW) (each infinite set owns a cluster point) and Borel-Lebesgue (BL) properties are equivalent, i.e. compactness and countably compactness are ...
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0answers
40 views

Definitions of proper maps

As far as I know, there several definitions of a proper map. A function $f\colon X\to Y$ is proper if it is continuous and for any space $Z$, the product $f\times \operatorname{id_Z}\colon X\times ...
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0answers
40 views

Give an example to show that $f_n$ fails to converge to $f$ uniformly over $S$ if $S$ is not compact

Given the theorem: Suppose $S \subset \Bbb R^n$ is compact, and $P$ is an equicontinuous sequence of functions ($f_n$) over $S$ converging pointwise to a function $f$ at each $x \in S$, then $f_n$ ...
2
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0answers
35 views

How to view Stone-Cech compactification of the real line?

I am going through Arveson's A Short Course on Spectral Theory and have come across an exercise constructing $\beta\mathbb{R}$ using the Gelfand map. I was wondering if there is an explicit ...
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1answer
34 views

Which subsets of $l^2$ are compact?

Let $$l^2=\left\{(x_n):\sum_{n=1}^{\infty}x_n^2<\infty\right\}$$ equipped with the norm $$\|(x_n)\|=\left(\sum_{n=1}^{\infty}x_n^2\right)^{1/2}.$$ State whether the following subsets ...
0
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1answer
51 views

Proof verification: Compact set has sup and inf

I was reading this post compact set always contains its supremum and infimum There was an answer reposted as follows: As $K$ is compact, we have that $K$ is bounded. So $\sup K$ and $\inf K$ ...
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1answer
34 views

Closed subsets of $\mathbb{C}^*$ proper for multiplication

Let $S_1$ and $S_2$ be two proper closed subsets of $\mathbb{C}^*$. Let's denote by $\overline{S_1}$ and $\overline{S_2}$ their closure in $\mathbb{C}_{\infty}.$ (Alexandrov compactification) ...
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0answers
25 views

Cantor-Bendixson rank of a first countable space

This question has been bothering me for quite a while, so let me ask it here. Is there a first-countable compact space $X$ with uncountable Cantor-Bendixson index? By a Cantor-Bendixson index I ...
2
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1answer
61 views

composition of functions is continuous

Question is as follows : Let $X,Y,Z$ are metric Spaces Let $f:X\rightarrow Y$ be continuous map onto $Y$ and let $X$ be compact. Also $g:Y\rightarrow Z$ such that $g\circ f:X\rightarrow Z$ is ...
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1answer
35 views

Is every compact metric space hereditarily separable?

Let $X$ be a compact metric space. I see why all open and closed subsets of $X$ are separable. But is every subset of $X$ necessarily separable? EDIT: Since $X$ is separable metric, it embeds into ...
0
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1answer
24 views

Metric spaces inside of metric spaces

Let $(X, d)$ be a metric space, $Y$ ⊂ $X$ and consider the metric space $(Y, d)$. Show that every open set $U$ in $Y$ has the form $U$ = $V$ ∩ $Y$ for an open set $V$ ⊂ $X$. Show that ...
1
vote
1answer
61 views

Is one-point compactification of a space metrizable

Let $X$ be a locally compact Hausdorff space.Let $Y$ be the one-point compactification of $X$. Two questions are: Is it true that if $X$ has a countable basis then $Y$ is metrizable? Is it true ...