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.

learn more… | top users | synonyms

8
votes
1answer
78 views

Motivation of paracompactness

"A paracompact space is a topological space in which every open cover admits a locally finite open refinement" is the definition of paracompactness on Wikipedia. Comparing with the definition of ...
1
vote
1answer
24 views

Second countable spaces under continuous closed surjective maps

Let $p:X\rightarrow Y$ be a closed continuous surjective map such that $p^{-1}(y)$ is compact for each $y\in Y$. Show that if $X$ is second countable then $Y$ is second countable. Let $y\in Y$ and $...
0
votes
1answer
24 views

A sequentially compact metric space is bounded. Help me fix this proof.

I know this is usually done by contradiction but I'm trying out something a bit different: Let $\mathbb{X}$ be a sequentially compact metric space. Let $(s_n)$ be a sequence in the metric space such ...
0
votes
0answers
33 views

The distance between disjoint closed sets may be zero [duplicate]

Let $K$ and $L$ be nonempty compact sets, and define $$d= \inf \{|x-y|: x \in K \wedge y \in L\}.$$ Show that it is possible to have $d = 0$ if we assume only that the disjoint sets $K$ and $L$ are ...
1
vote
1answer
26 views

Counterexample of intersection characterization of compactness

Given metric space ($X,d$), let {$S_1, S_2,...$} be a set of non-empty sets where $S_1 \supseteq S_2...$, then if $X$ is compact and the $S_t$ are closed then $ \cap_tS_t$ is not empty. In $\Bbb R$, ...
1
vote
1answer
45 views

Is the fact true that compact set is in perfect set?

By definition, Perfect set $E_1$ is closed set without isolated points. Compact set $E_2$ is bounded and closed set in Euclidean space; $\mathbb{R}^n$. Is the following equation true? $$E_2 ...
1
vote
1answer
32 views

Definition request: explicit definition of covering compactness in terms of set notation

Part of my confusion with covering compactness stems from the fact that it is a definition given almost completely in a high level manner (in English no less). When I look at: A set $A \subset (...
2
votes
2answers
67 views

Show that $\int_{a}^{b}{x^{n}f(x)dx}=0$, then $f=0$

Let $f:([a,b],\vert\vert)\to (\mathbb{R},\vert\vert)$ a continuous function. Show that, if $$\int_{a}^{b}{f(x)x^{n}dx}=0$$ for all $n\in\mathbb{N},n\geq 0$, then f is identically zero. My attempt: ...
-1
votes
2answers
62 views

What does “closed” mean in Heine Borel for $C^0$?

Heine Borel for $C^0$: A set $\mathcal{E} \subseteq C^0([a,b], \mathbb{R})$ is compact if it is closed, bounded and equicontinuous. I don't really understand what closed mean in the definition....
6
votes
2answers
82 views

For a compact metric space $X ,f: X \rightarrow X $ s.t $ d(x,y)\leq d(f(x),f(y))$ is surjective [duplicate]

I want to show that in a compact metrix space $X$ ,the function $f :X \to X$ such that $d(x,y) \le d(f(x),f(y))$ is surjective! I tried to show that f is continuous and injective but i don't think it ...
7
votes
4answers
987 views

What is wrong with this argument that closed interval [0, 1] is not compact?

I am a student majoring engineering. I am studying real analysis with textbook 'Measure and Integral' by Wheeden and Zygmund. This book defined compact like the following: $E$ is compact if ...
2
votes
2answers
56 views

An application of Urysohn's lemma

Let $X$ be a Compact Hausdorff space. Assume that the vector space of real valued continuous functions on $X$ is finite dimensional. Show that $X$ is finite. Suppose $X$ is infinite, given any $n\in \...
1
vote
1answer
56 views

Proving that continuous image of a closed bounded subset of $\mathbb{R}$ is closed and bounded (without compactness)

I wish to prove that continuous image of a closed bounded subset of $\mathbb{R}$ is closed and bounded (the function is $\mathbb{R}\rightarrow\mathbb{R} $). However, I do not have the equivalence to ...
1
vote
1answer
25 views

Graph of function is compact

Let $X$ be a Hausdorff space. Let $f:X\rightarrow \mathbb{R}$ be such that $\{(x,f(x)):x\in X\}$ is a compact subset of $X\times\mathbb{R}$. Show that $f$ is continuous. What i have done so far is : ...
0
votes
1answer
24 views

Which statement is true for non-compact sets in a metric space [closed]

We know that a set is compact if for every open cover, there exists a finite subcover. If a set is not compact then is it true that: There exists an open cover, such that there does not exists a ...
0
votes
1answer
25 views

Compact Metric Spaces & Triangle Inequality Theorem [duplicate]

Let X be a metric space, p ∈ X, and let K ⊂ X be compact. Show that there exist x0, x1 ∈ K such that d(x0, p) ≤ d(x, p), ∀ x ∈ K, d(x1, p) ≥ d(x, p), ∀ x ∈ K. I know that I have to show the distance ...
1
vote
2answers
44 views

X compact iff projection is closed

I've seen this exercise around to motivate the definition for complete variety, but I seem to have trouble proving it (and can't find any hints). The statement I want to show is: $X$ is compact if ...
0
votes
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 e_{n}-...
3
votes
1answer
58 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
vote
1answer
40 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
votes
2answers
54 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
votes
0answers
49 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) =...
0
votes
0answers
81 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 $f$ ...
1
vote
2answers
43 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
votes
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
vote
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
vote
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
votes
1answer
64 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
votes
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
vote
1answer
41 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 ...
0
votes
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
votes
1answer
36 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
vote
1answer
35 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
votes
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 $\{(t-\...
0
votes
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
votes
0answers
65 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 \...
1
vote
3answers
38 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
votes
1answer
40 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
votes
1answer
46 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 ...
1
vote
1answer
47 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
vote
1answer
48 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 ...
9
votes
2answers
148 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
votes
2answers
23 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
votes
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. $t^...
3
votes
1answer
48 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
votes
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 ...
7
votes
4answers
221 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
votes
3answers
50 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
votes
2answers
70 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 $0$...
3
votes
3answers
75 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 ...