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

6
votes
2answers
513 views

What's wrong with this 'open cover' of the Koch Snowflake?

This question is to help me find peace. First, the question of the Snowflake's compactness has been tackled here on this site: Is the Koch Snowflake a Compact Space? Is Koch snowflake a continuous ...
2
votes
1answer
195 views

Connected and Compact preserving function is not continuous example?

Before we start, I'm aware the result is true for when the function is a map between Euclidean spaces. In fact, with a minimal amount of extra work we can see that a function between locally-compact, ...
2
votes
0answers
185 views

Counterexample to Converse of Extreme Value Theorem?

The extreme value theorem says: If $X$ is a compact topological space, then for all functions $f: X \to \mathbb{R}$ such that $f$ is continuous we have that $f$ satisfies the extreme value property. ...
1
vote
0answers
32 views

Compactness in Sobolev spaces

I am looking for characterizations of compactness in the Sobolev space $H^{-1}$. In particular, I am looking for a characterization involving the Fourier transform. Can anyone suggest some results ...
0
votes
2answers
136 views

a compact set $X$ has a countable set $S$ such that $\overline{S} = X$

Suppose $X \subseteq \mathbb{R}^d$. Suppose $X$ is compact. Then there exists a countable subset of $X$, $S \subseteq X$ such that $\overline{S} = X$. How can I show this? I have no idea how to ...
2
votes
0answers
91 views

Prove that every pseudocompact metric space is compact

This is from Real Mathematical Analysis by Pugh, problem 2.85(a). I've seen proofs but they've used concepts that haven't been covered up to this point, like the Tietze extension theorem, metrizable ...
18
votes
2answers
1k 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 ...
3
votes
2answers
82 views

Proving the set $C = \{\,x \in \mathbb R^n : \sum x_i = 1, x_i \in [0,1]\,\}$ is compact.

Proving the set $C = \{\,x \in \mathbb R^n : \sum_{1}^n x_i = 1, x_i \in [0,1]\,\} \subseteq \mathbb R^n$ is compact. Alright: I can use the Heine-Borel theorem to prove this, therefore all I need to ...
2
votes
2answers
113 views

Compact Domain and Inverse Image

I am trying to show that given $f:M \rightarrow N$, where $M$ is compact, $f$ is continuous and onto, then given $A \subset N$: $$ f^{-1}(A) \text{ closed} \implies A\text{ closed} $$ I am dealing ...
0
votes
1answer
85 views

Open, closed, bounded or sequentially compact

How do I find if this set is open, closed, bounded or sequentially compact? $$S=\left\{z:5<\left|z\right|\leq7\right\}$$ I find the value of $z$ is: $-7\le z < 0$. Can you please explain. Thank ...
0
votes
0answers
63 views

Equivalence conditions in the Heine-Borel theorem for the real line

The Heine Borel theorem (book, pg 335) shows that the following conditions are equivalent- A set $K$ is closed and bounded. $K$ is compact. My question is that in the proof of 1 $\implies$ 2 where ...
1
vote
2answers
325 views

Compactness implies Continuity?

I am stuck on this question (probably there are many counterexamples, but I can't find any). "Suppose $f:\mathbb{R}\mapsto\mathbb{R}$ that preserves compactness (i.e, for every $K \subseteq R$, then ...
0
votes
1answer
85 views

A counterexample on compactness (closed vs complete)

In a metric space $M$: If $A \subset M$ is complete and for each $\epsilon > 0$ there exists a compact $K \subset M$ with $A \subset \{ x \in M : d_M(x, K) \leq \epsilon \}$ then $A$ is compact. ...
1
vote
1answer
137 views

Locally-compact function spaces?

I ask this question out of curiosity, not a specific need. Euclidean spaces and manifolds. Are there examples of locally compact function spaces? Could (some?) Sobolev spaces be locally compact?
3
votes
2answers
88 views

Čech-Stone compactification of $\mathbb N$ and ultrafilters on $\mathbb N$

I have found in the literature that the Čech-Stone compactification $\beta\mathbb N$ of $\mathbb N$ (or more generally, of any discrete topological space) can be identified with ultrafilters on ...
1
vote
1answer
42 views

How can a bounded subspace of the left order topology be compact?

I want to show that every bounded set equipped with the left order topology is compact. This is a statement I found on a wikipedia page and appearently it is lifted from the book Counterexamples in ...
3
votes
2answers
100 views

Why is the image of a compact operator separable?

Let $A$ and $B$ be normed vector spaces and let $S\in \mathscr{K}(A,B)$ be a compact operator. Question: How does it follow that the image of $S$ is separable? Thanks for the help.
3
votes
1answer
49 views

Analysis question.

Is this set compact? $\{(x,y) \in \mathbb R^2 : |x|+|y|\leq 1\}$. I know that is closed and bounded so compact but I don't know how to show it is closed and bounded mathematically. This is the graph ...
0
votes
1answer
62 views

Compact features

Consider this problem: Let $X$ be a metric space, $U$ be open, $K$ compact and $K\subset U$, show that there exists a $r>0$ such that $B(k,r)\subset U$ $\forall k\in K$ Here $B(k,r)=\{x\in X ...
4
votes
3answers
348 views

Proof help. Core-compactness, Hausdorff, Locally Compact

While reading about topologies on continuous function spaces, I've seen remarks that core-compact and locally compact are equivalent for Hausdorff spaces. Now I can clearly see that locally compact ...
2
votes
4answers
220 views

Why is $C_c^\infty(\Omega)$ not a normed space?

I am watching a Coursera video on Théorie des Distributions and I am trying to understand one of the slides. Let $\Omega \subset \mathbb{R}^N$ be an open set and $C_K^\infty(\Omega) = \{ \phi \in ...
2
votes
0answers
37 views

Compactness in topology of uniform conergence (of functions and all their derivatives) on compact subsets of (0,\infty)

I am trying to understand an example in the book "Lectures on Choquet's Theorem" (R.R. Phelps). My question is: Given the space of real valued infinitely differentiable functions on $(0, \infty)$ ...
2
votes
1answer
164 views

Sequence of elements having a convergent subsequence -NBHM $2014$

Question is to find which of the following are true? Let $V$ be the space of continuous functions on $\mathbb{R}$ with compact support endowed with metric ...
0
votes
1answer
70 views

Noncompactness of the closed unit ball in $L^2$

Let $$ L^2[0,1]=\{f:[0,1]\to\mathbb R\,\,\text{such that}\,\, \|f\|_2<∞\}, $$ where $\|f\|_2^2=\int_0^1 |f(x)|^2\,dx.$ Show that the unit sphere $$ S=\{f\in L^2[0,1]:\|f\|_2\le 1\}, $$ is ...
5
votes
2answers
81 views

Connectedness and compactness of a union of two sets

Let: $$A=\Big\{ (x,y) \in \mathbb R^2: 0 \le x \le 1, y=\frac{x-1}{n},\, n\in \mathbb N \Big\}$$ $$B=\Big\{ (x,y) \in \Bbb R^2: 0 \le x \le 1, y=\frac{x}{n},\, n\in \mathbb N \Big\}$$ Is $A \cup B$ ...
1
vote
1answer
51 views

One-point compactification of the union of circle and an intersecting open interval.

I have to give the one-point compactification of $S^1 \cup \{(0,2) \times \{0\}\}$. I think I can see this as a circle with two 'tails' with open ends, one on the inside, one on the outside. Is this ...
1
vote
0answers
57 views

$\sigma$-$\sigma$-compactness is $\sigma$-compactness?

I mean, if $X=\displaystyle\bigcup_{n\in\mathbb{N}}K_n$ where each $K_n$ is $\sigma$-compact, then $X$ is $\sigma$-compact? I'm not sure if a countable union of countable unions is still a countable ...
1
vote
2answers
83 views

Is the following set is compact

Consider the set of all $n \times n$ matrices with determinant equal to one in the space of $\mathbb R^{n\times n}$. My idea is compact because determinant function is continous ant it is bijective ...
1
vote
1answer
38 views

Show that $A$ is non-compact

I have a problem: For $C\left [ 0,1 \right ]=\left \{ x:\left [ 0,1 \right ] \to \Bbb R \ \text{is continuous on } \left [ 0,1 \right ] \right \}$, with a norm: $$\left \| x \right \|=\sup_{t\in ...
2
votes
0answers
43 views

Is an ideal generated by a compact subset finitely generated?

Let $R$ be a commutative topological ring and let $K$ be a compact subset of $R$. Denote by $I$ the ideal generated by $R$. Then is it true (or under what assumptions on $R$ (besides Noethernity)) is ...
1
vote
2answers
130 views

Characterizing $\sigma$-compactness via closed sets

A topological space that is a countable union of compact subsets is called $\sigma$-compact. My intuition says that the follow property should be equivalent to $\sigma$-compactness: $$ \text{Every ...
6
votes
4answers
276 views

Which of the following subsets of $M_n(\mathbb{R})$ are compact (NBHM)

Following is a list of problems from an exam for admission into Ph. D program. I have just compiled all previous questions on compactness of certain subsets of matrices and i tried to work out . I ...
2
votes
1answer
127 views

Prove that the countable complement topology is not meta compact?

I have seen the proof of (countable complement topology is not meta compact) , which says that the countable intersection of open sets is open and thus uncountable, so this topology cannot be meta ...
2
votes
1answer
638 views

In a Hilbert space, every bounded and closed set is weakly relatively compact.

My aim is to prove that in a Hilbert space, any sequence has a weakly convergent subsequence. To prove this, I'm trying to prove that: ...
3
votes
2answers
251 views

Any ball is connected?

Let $X$ be a compact , metric space. Assume that the closure of every each open ball it the closed ball with same center and radius. Prove that any ball in this space is connected.
5
votes
2answers
133 views

On the compacity of the space of probability measures

Let $X$ be a complete metric space, and denote by $\mathcal{P}(X)$ the set of probability measures on $X$. I am interested in proving that if $X$ is compact then $ \mathcal{P}(X)$ must be compact in ...
4
votes
1answer
55 views

Topological space in which there are no close and compacts subsets (except for the empty set)

Any example of those topological spaces? I cant think of no one :S I think it must be infinite and it must not be T2, but no idea how to find one.
4
votes
1answer
67 views

Suppose that for all $t <1$ there are points $x_t$ and $y_t$ such that $d(x_t,y_t) = t$.

Let $(X,d)$ be a compact metric space. Suppose that for all $t <1$ there are points $x_t$ and $y_t$ such that $d(x_t,y_t) = t$. Prove that there exists points $x$ and $y$ such that $d(x,y) = 1$. I ...
2
votes
1answer
84 views

Hilbert space, orthonormal system, compact set of vectors

Could you help me solve this problem? Let $e_1, e_2, ...$ be an orthonormal system in a Hilbert space, $\delta_1, \delta_2 ... \in (0, + \infty)$. Prove that the set of all vectors $\sum _{n=1} ...
2
votes
1answer
59 views

Compactness in $\mathbb{R}^{X}$

I'm reading a book chapter on weak topology, where the author identified the collection of all real functions on an abstract space $X$ with $\mathbb{R}^{X}$. I find it difficult to make sense out of ...
1
vote
2answers
262 views

Prove that :- If K is a compact subset of R with non empty interior then it is of the form [a,b] or [a,b] - U{In}

The question is :- Let $K$ be a compact subset of $\mathbb R$ with non empty interior. Show that K is of the form $[a,b]$ or $[a,b] \setminus \bigcup I_n$ , where { $I_n$} is a countale disjoint ...
6
votes
1answer
241 views

Weak Hausdorff space not KC

I am stuck with a problem in general topology. First of all, recall that a space $X$ is KC if every compact subset of $X$ is closed, and is weak Hausdorff if for all $u:K\rightarrow X$ continuous ...
-1
votes
1answer
58 views

**Are there infinite metric spaces which have no infinite compact subsets**

Please if anyone can help me in solving this example: Are there infinite metric spaces which have no infinite compact subsets If there are, please, that they look like, please help me, thank you, ...
5
votes
6answers
2k 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
1answer
183 views

Compact sets in R

Prove that a compact set $K$ in $R$ is of the form $[a,b]$ or $[a,b]$ - $X$ where $X$ is countable union of disjoint open sets such that the end points are in K.
0
votes
3answers
90 views

closed and bounded but not compact set of real-valued bounded functions

I'm trying out a problem I was given and this is the statement: Prove, or disprove, that every bounded and closed subset of the set of real-valued and bounded functions on [0,1] equipped with the sup ...
4
votes
1answer
199 views

Stone-Čech compactification of discrete space

Let $X$ be a discrete space and $\beta X$ its Stone-Čech-compactification, given by $\overline{\iota(X)}$ where $$ \iota:\ X \to \prod_{f \in C(X,[0,1])} [0,1],\quad x \mapsto (f(x))_{f \in ...
0
votes
3answers
67 views

Is the $\omega$-product of the set of irrationals compact?

We know that any product of compact spaces is compact. But, I wonder that the countable product of $\mathbb{P}$ can be compact since $\mathbb{P}$ is not compact?
0
votes
3answers
62 views

Compact subsets of a metric space

I am trying to to prove that f: X --> Y is continuous on X if and only if f is continuous on every compact subset of X. X and Y are metric spaces. How do I show that every point of X belongs to some ...
0
votes
1answer
324 views

Proving that a Closed Interval Is Compact

My text (Stoll, Introduction to Real Analysis, 2nd Ed) defined that $K$, a subset of $\mathbb R$, is compact if every open cover of $K$ has a finite subcover of $K$. Then, it proceeded to prove that ...