Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [compactness]

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.

0
votes
0answers
12 views

Show that $K∩L$ is homeomorphic to $A$, where $K ∩ L = \{\{(a,0),(a,1)\} : a \in A\}$ is a subset of equivalence classes in $[0,1] \times \{0,1\}$.

I'm trying to solve the following question: Let $X$ be a topological space and $A$ a subset of $X$. On $X\times\{0,1\}$ define the partition composed of the pairs $\{(a,0),(a,1)\}$ for $a\in A$, ...
1
vote
1answer
18 views

Metric spaces problem regarding if composition function is uniformly continuous then what about individual function itself

Suppose $X, Y, Z$ are metric spaces and $Y$ is compact. Let $f$ maps $X$ into $Y$. Let $g$ be continuous one-to-one map $Y$ into $Z$ and put $h(x)=g(f(x))$ for $x$ in $X$. If $h(x)$ is uniformly ...
3
votes
1answer
37 views

Are neural networks with bounded parameters a compact subset of the Banach space of continuous functions?

Let $d, n \in \mathbb{N}$. Moreover, let $D \subset \mathbb{R}^d$ be compact and denote with $\mathcal{C}(D, \mathbb{R}^n) $ the set of continuous functions from $D$ to $\mathbb{R}^n$. Then $\mathcal{...
0
votes
0answers
23 views

The two statements about compactness [on hold]

In the textbook Understanding Analysis by Stephen Abbot, it defines a set K in R is compact if every sequence in K has a subsequence that converges to a limit that is also in K When I watched a ...
4
votes
1answer
44 views

Proving that a subset of $l^{2}$ is compact

Define $B \subset \ell^{2}$ by $$B = \{x \in \ell^{2}: \sum_{n=1}^{\infty}n|x_{n}|^{2} \leq 1 \}. $$ Show that $B$ is compact. I found this question while studying for an exam. I tried proving that $...
15
votes
4answers
3k views

Why are subsets of compact sets not compact?

So much of the properties of compact sets are motivated by finite sets, to the point that thinking of compact sets as topologically finite sets may yield some deeper understanding. But finite sets ...
0
votes
0answers
38 views

Given $E \subset \mathbb{R^n}$ with $m^*(E) < \infty$. Show that if $E$ is compact then $m^*(E) = \lim_{m \to \infty} m^*(\sigma_m)$

So, here is the formal statement: Let $m^*$ denote the Lebesgue outer measure on $\mathbb{R^n}$, and suppose $E \subset \mathbb{R^n}$ with $m^*(E) < \infty$. Let $\sigma_m = \{x\in \mathbb{R^n} :...
1
vote
3answers
40 views

Compact or open $\{0\}\cup\{\frac1n + \frac1m | m,n \in N\})$ in R?

Compact or open $\{0\}\cup\{\frac1n + \frac1m | m,n \in N\})$ in R ? The question is straight forward There exists no interval about $2\in S$ that has only elements of S. Not open What about ...
0
votes
2answers
47 views

How do I know $x \in \bigcap\limits_{n = 1}^\infty f^{-1}([n,\infty))$?

Suppose $X$ is a compact metric space and $f: X \rightarrow \mathbb{R}$ is a function for which $f^{-1}([t,\infty))$ is closed for any real $t$. Then $f$ achieves its maximum value on $X$. I am ...
0
votes
1answer
16 views

Min over Dense Set

Suppose that $(X,d)$ is a compact metric space, $D\subseteq X$ is a dense subset, and $f:X\rightarrow (-\infty,\infty]$ is a proper, lsc, convex functional. Then is it true that $\inf_{\tilde{y} \in ...
1
vote
1answer
37 views

Proof Clarification: Show that every sequence of a compact set $S \subset \mathbb{R}$ has a subsequence which converges in $S$.

So, my textbook gives a proof for the following statement Every sequence of a compact set $S \subset \mathbb{R}$ has a subsequence which converges in $S$. Textbook Proof: Assume that every open ...
0
votes
1answer
35 views

Theorem 3.20 rudin's functional analysis, compactness of $K = f(S \times A)$

Reading through theorem 3.20, Rudin's functional analysis (point (a)). If $A_1,\ldots, A_n$ are compact convex sets in a topological vector space $X$ then $co(A_1 \cup \ldots \cup A_n)$ is compact. ...
3
votes
2answers
45 views

Non compacity of a subset of $\ell^1$

Consider $l^1 = \left\{ \{a_n\}_{n=1}^\infty : a_n \in \mathbb{R}, \ \ \sum_{n=1}^\infty |a_n| < \infty \right\}$ and let $K = \left\{f \in l^1 : |f(k)| \leq \frac{1}{k} \ \ \forall k\right\}$. ...
4
votes
0answers
31 views

Is $ \left\{\cup _{n \in \mathbb{N} } (x, x^n) \mid x \in [0,1] \right\} $ compact?

Is $$ \left\{\cup _{n \in \mathbb{N} } (x, x^n) \mid x \in [0,1] \right\} $$ compact? My answer would be it isn't because it doesn't contain all its accumulation points ( for example the point $ (1/...
0
votes
1answer
39 views

Homeomorphism from punctured sphere to horn torus

Im working on a problem about 1 point compactification, and i am at a step where I want to take the punctured sphere $$ S^{2}\setminus\left\{ (0,0,1)\right\}= \left \{ (x,y,z)\in\mathbb{R}^3 \, \mid \,...
1
vote
3answers
60 views

Does Cauchy Completeness imply the Heine-Borel theorem generally?

I've been working through some reverse math with the completeness definitions of a metric space. More over, I've learned that in a metric space X that is ordered, The Least Upper Bound Property, ...
2
votes
1answer
32 views

Questions about compact manifolds

I have two question. Let $M$ and $N$ two compact manifolds. 1) It is true that $C^{\infty}(M\times N)\cong C^{\infty}(M)\otimes C^{\infty}(N)$??. 2) Taking $f$ $\in$ $\Omega^1(M)=\Gamma(T^{\ast}M)$ ...
2
votes
1answer
63 views

Nested sequence of compact subsets covering an open set in $\mathbb{C}$

Let $U$ be an open set in $\mathbb{C}$. I would like to prove the following result: There exists a sequence of compact sets $\{K_n\}$ with the following properties: Each $K_n$ is a subset of $U$. $...
0
votes
1answer
32 views

Notation for compact sets?

Is there some accepted notation for a set that is compact? E.g. I am currently writing "... [blah] is true if for every compact set $A \subset \mathbb{R}^n$ and ...". I could simplify my writing if ...
0
votes
2answers
70 views

Rudin Analysis, Theorem 2.36: Is there a generalization?

In Rudin's Mathematical Analysis, he states theorem a theorem for compact sets. If $\{K_\alpha\}$ is a collection of compact subsets of a metric space $X$ such that the intersection of every ...
2
votes
1answer
51 views

A self-adjoint operator with essential spectrum={0} is compact

Does every self adjoint operator (on a Hilbert space) with essential spectrum={0} is a compact operator ?
0
votes
2answers
18 views

Behaviour of positive function on compact sets

Let $v$ be any strictly positive bounded function on $\Omega \subset \mathbb{R}^n$ and zero on $\partial \Omega.$ Can we say that for each compact subsets (w.r.t usual topology defined on $\mathbb{R}...
8
votes
1answer
550 views

Why don't we use closed covers to define compactness of metric space?

I'm a beginner in metric space. So many books I've read, there is only the notion of open covers. I want to know why do we worry about open covers to define the compactness of metric spaces and why ...
2
votes
1answer
35 views

Weak derivative of a Sobolev function in unbounded domains

I have following setup: $f\in W^{1,1}(\mathbb{R}^3)$ with $f\geq 0$ and $\int_{\mathbb{R}^3}f=1$, how can I show that the weak derivative of $\tilde{f}(x):=\int_{\mathbb{R}^2}f(x,y,z)\,\text{d}y\text{...
0
votes
2answers
54 views

General topology, compact sets, neighborhoods

Consider the following statements.If a statement is true supply a proof; if false give a counterexample. (a) Two distinct points p and q can be separated by disjoint neighborhoods. (b) A set K = {p1, ...
-1
votes
1answer
43 views

Proof that a set is closed, having compactness

8. Let $G$ be a non-empty open subset of $\mathbb R$. Let $x_0\in G$ and put $F=[x_0,\infty) \cap G^c$. Here $[x_0,\infty) = \{x\in \mathbb R : x\ge x_0 \}$. (a) Prove that $F$ is a non-empty set. ...
0
votes
1answer
36 views

Prove: [0,1] is supercompact

Question 7.2.16 from S. Morris's Topology without Tears I'm just trying to get a grip on how these concepts work. So I've written this up for checking. Prove that $[0,1]$ with the euclidean ...
0
votes
1answer
29 views

Properties of this topology on $\Bbb X$.

For given the usual topology $\tau$ on $\Bbb{R}$, define the compact complement topology on $\mathbb{R}$ to be $$\tau'=\{A\subseteq \Bbb{R}:A^C\text{ is compact in }(\Bbb{R},\tau)\} \cup \{\emptyset \...
0
votes
1answer
29 views

General topology, compact sets, neighborhood

I'm really struggling in writing the proof of these statements My answers are: 1)true 2)true 3)false but i can't supply the proof for this. Any help please? I don't know how to write proofs. I'm new ...
0
votes
3answers
44 views

Properties of this topology on $\mathbb N$.

$\mathbb N$ is consist of the basis generated by the set $A_n=\{n,n+1,n+2,n+3....\}$ . Then what properties does it have? Hausdorff: for any natural numbers $x$ and $y$ there are no disjoint open ...
0
votes
0answers
31 views

Why the given groups are compact?

I was told that $Z_{m}$ is a compact topological group. But I do not know why it is compact ..... could anyone explain this for me please?
1
vote
1answer
23 views

If $K_1$ & $K_2$ are disjoint nonempty compact sets ,show that there exist $k_i$ $\in$ $K_i$

If $K_1$ & $K_2$ are disjoint nonempty compact sets ,show that there exist $k_i$ $\in$ $K_i$ such that $|k_1 - k_2|$=inf{$|x_1 - x_2|$: $x_i$ $\in$ $K_i$}. They are all subsets of $\mathbb ...
0
votes
1answer
16 views

A characteristic continuous mapping

A mapping $f : X \rightarrow Y$ is said to be $characteristic$ if for every compact $C \subseteq Y$ the preimage $f^{-1}(C) \subseteq X$ is also compact. Let $f : \mathbb{R} \rightarrow \mathbb{R}$ ...
-2
votes
2answers
81 views

Is set $[0,1)$ compact in $\mathbb{R}$? [closed]

I know that $[0,1]$ is compact in $\mathbb{R}$ using nested interval theorem. But I am stuck for the case of $[0,1)$. How to do this?
0
votes
2answers
68 views

Discreteness“+” Compactness= Finiteness: What does this really mean?

In my journey of self-studying topology, I have always come here to gain some insights on topics that really didn't click for me, and it's marvelous to have such a place where you can come back when ...
0
votes
0answers
31 views

If $\beta$ is a continuous function with $\beta f'=0$ for all $f\in C_c^\infty$, are we able to conclude $\beta=0$?

Let $\beta:\mathbb R\to\mathbb R$ be continuous. If $$\beta f'=0\;\;\;\text{for all }f\in C_c^\infty(\mathbb R),\tag1$$ are we able to conclude $\beta=0$? If, given a compact $K\subseteq\mathbb R$, ...
0
votes
1answer
46 views

Is there a counterexample such that : $X$ is a topology space , every infinite subset of $X$ has a limit point but $X$ is not compact .

$X$ is a topology space and $A=\{x_1 , x_2 , ... , x_n , ... \}$ is a subset of $X$ , $y$ is a limit point of $A$ . Can we show that for each $N$ , $y$ is also a limit point of $A'=\{x_N ,x_{N+1}, .....
1
vote
0answers
43 views

An alternative way to show that any two norms on a finite dimensional vector space are equivalent. [closed]

I encountered this different method on page 432 of John Lee's Introduction to Smooth Manifolds. The hint in the book states that first choose an inner product on the vector space, and show that the ...
0
votes
1answer
57 views

Product of a compact topological space and a singleton in another topological space is compact proof

It's before we prove that 'Product of two compact sets is compact'. There are topological spaces $X$(which is compact), $Y$ and the product topology on $X \times Y$ is given by the subbase $U \times V$...
1
vote
1answer
21 views

Continuous maps between compact Hausdorff spaces and their induced maps on their space of continuous real-valued functions.

Let $X,Y$ be two compact Hausdorff spaces and let $ \alpha: X \to Y$ be continuous onto map. Let $\alpha^{*}$ be the induced map from $C(Y)$ to $C(X)$ by mapping any $f \in C(Y)$ to $f \circ \alpha \...
0
votes
2answers
37 views

Product of a compact set and a singleton is compact proof

It's before we prove that 'Product of two compact sets is compact'. Let $S$ be an open cover of $X \times \{\bullet\}$ where $X$ is compact. Then $\pi_1(S)$ is an open cover of $X$ so there is a ...
0
votes
2answers
37 views

Compactness of a closed subspace

I was looking at the following proposition : "Every closed subspace of a compact topological space is compact" and I am wondering why the following proof is not good : Let $(X, \tau)$ be a compact ...
0
votes
1answer
26 views

If $A$ and $B$ are closed, then $A+B$ is Borel

I am trying to prove this statement by proving first that $A+B = \{a+b : a \in A, b \in B\}$ is $F_\sigma$. I am lost, I saw that I had to prove first the case when $A$ and $B$ were compact then $A+...
1
vote
1answer
55 views

Compactness and connectedness in $\Bbb R^3$

Consider the set $$A=\left\{ \begin{pmatrix} x\\y\\z\end{pmatrix} \in \Bbb R^3: z=x^2+y^2+1\right\} \subset \Bbb R^3$$ Prove of disprove: $A$ is connected and compact The set $A$ is unbounded, ...
1
vote
1answer
44 views

Have I understood Compact Set correctly

In our current Measure Theory Class, we bought up the notion for a function $f:\mathbb R \to \mathbb R$ that is continuous to have a compact support, is equivalent to the fact that $\overline{\{x \in \...
0
votes
1answer
21 views

Continuous function on compact metric space has minimum and maximum

Let's have continuous function $F$, $B$ - closed ball around some point $x_0$ with radius $\rho$, and some set $J$. $$\max_{t∈J,x∈B(x_0,ρ)}||F(x,t)||$$ We have one theorem which says that continuous ...
0
votes
4answers
40 views

Strategy to prove a set is compact. Case: $S=\{0\}\cup\bigcup_\limits{i=1}^{\infty}\{\frac{1}{n}\}$ [duplicate]

Verify that $S=\{0\}\cup\bigcup_\limits{i=1}^{\infty}\{\frac{1}{n}\}$ is compact subset of $\mathbb{R}$ while $\bigcup_\limits{i=1}^{\infty}\{\frac{1}{n}\}$ is not. If I take the sequences $\{\frac{1}...
3
votes
1answer
50 views

Showing that $f:M\mapsto M$ has a unique fixed point if $M$ is compact and $d(f(a),f(b))<d(a,b)~\forall a,b\in M$

I'm not sure if this is a good approach (I know there is a different proof that involves a function $a\mapsto d(a,f(a))$ whose minimum is the unique fixed point): For any $a\ne b\in M$ we have that $...
0
votes
1answer
42 views

Isometry $f:M\mapsto M$ is surjective if $M$ is compact: proof by composing $f$ with itself

Let $(M,d)$ be a compact space and $f:M\mapsto M$ an isometry. To prove that $f$ is surjective we take an arbitrary $a\in M$ and consider $f^n:M\mapsto M$ the function obtained by composing $f$ $n$ ...
1
vote
1answer
33 views

Does adding a continuous inequality constraint over a compact set lead to another compact set?

so the problem is as follows. I have the vector space $x=[x_1,x_2,...x_N] \subseteq R^N, 0 \leq x_{1,2,...,N} \leq M$ and I extract from it a subset by adding a constraint of this kind: $X^1=\{x \...