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. This includes logical compactness.

Filter by
Sorted by
Tagged with
1
vote
2answers
53 views

Uncountable product of T1 sequentially compact spaces is never sequentially compact (continuum hypothesis)

How to prove the statement: Assuming the continuum hypothesis, the product of any uncountable family of $T_1$ spaces, each having more than one point, is never sequentially compact. The statement ...
2
votes
1answer
63 views

Compactness theorem equivalences

i have this equivalence to compactness theorem that i have problems to prove: For every first-order theory $T$, every tuple $x̄$ of distinct variables and all sets $\Phi(x̄),\Psi(x̄)$ of first-order ...
-1
votes
0answers
17 views

Functional Analysis : completion of normed spaces

The question says: Let be $N$ a normed space with $\dim(N)<\infty$. Suppose that exists a subset $X$ of $N$ such that $X$ has an opened subset $U$. Show that $X$ isn't compact. This question is ...
2
votes
1answer
62 views

Characterization of continuity via closed.

Let $K \subset\mathbb R^{n}$ a compact set and $f : K \rightarrow\mathbb R^{m}$ a continuous and one-to-one function. Show that the function $f^{-1} : f(K) \rightarrow K$ it is continuous. Hint: By ...
0
votes
1answer
25 views

Compact Embedding of Sobolev Valued Functions

I am wondering if it is possible to show $L^{2}((0,T),H^{1}(\Omega))$ is compactly embedded in $L^{2}((0,T),L^{2}(\Omega))$. It's certainly true that $H^{1}(\Omega)$ is compactly embedded in $L^{2}(\...
2
votes
1answer
72 views

In metric space, countable union of compact sets is separable

I'm trying to prove the next statement: If $(X,d)$ is a metric space and $K_n\subseteq X$ a compact sub-set for every $n\in\mathbb{N}$ then: $\bigcup_{n=1}^{\infty}K_n$ is separable. In my attempt I ...
0
votes
1answer
27 views

Is the following an open cover for the set $K = \{0\}\cup \{1/n\mid n \in \mathbb{N}\}$?

I know it has been asked to death on this site how to prove that $K = \{0\}\cup \{1/n\mid n \in \mathbb{N}\}$ is compact. I would like to not be spoiled about the proof as IMO my question is ...
1
vote
0answers
25 views

Properties of the space of Radon measures

Let $\mathcal{M}(X)$ be the space of Radon Measures on $X$ and let be $\mu \in \mathcal{M}(X)$, $ \mu$ absolutely continuous w.r.t Lebesgue Measure with densitity $f.$ If $X$ is a compact how can I ...
0
votes
1answer
25 views

Clarification on the theorem 2.33 of Baby Rudin: If $K \subset Y \subset X$, then $K$ is compact relative to $Y$ iff $K$ is compact relative to $X$

The theorem is that: Suppose $K \subset Y \subset X$. Then $K$ is compact relative to $X$ if and only if $K$ is compact relative to $Y$. Rudin begins the proof by stating that: Suppose $K$ is a ...
-2
votes
0answers
43 views

Is $f(x,y,t)= tx +( 1 - t)y$ continuous? [closed]

If $K$ is compact in $\mathbb R ^{n}$ and $f\colon K\times K \times [0,1] \rightarrow \mathbb R^n$ such that $f(x,y,t)= tx +( 1 - t)y \in \mathbb R ^{n}$ with $x,y\in K $ and $t \in [0,1]$ Show that $...
1
vote
1answer
20 views

metric for the quotient space $S^n \otimes S^n \otimes \ldots \otimes S^n /S_m$

Here $S^n $ denotes the unit sphere in an $n$-dimensional (complex) Hilbert space. We have $m$ copies of this sphere. By $S_m $ we mean the permutation group of order $m$. By the quotient, we identify ...
0
votes
0answers
16 views

A compact operator in Hilbert space is the limit of a finite rank ssequence of operators [duplicate]

If $H$ is a hilbert space and $T\in K(H)$. Then there is a sequence of finite rank operators {$T_k$} which converges to $T$ in $B(H)$. By a claim we have that for every bounded sequence $(x_n)$ in $H$,...
0
votes
1answer
28 views

What do we know about the set AB if A en B non-empty subsets of $\mathbb{R}_0^+$ with different conditions on A and B

If $A$ en $B$ non-empty subsets of $\mathbb{R}_0^+$ and we say that $AB=\{ ab| a \in A $ and $ b \in B \}$. a)If $A$ and $B$ are open, $AB$ open? I thought this was true. I wanted to proof if there ...
3
votes
2answers
53 views

Collection of all compact subsets of a Hausdorff space $X$ is compact if and only if $X$ is compact.

Let $X$ be a Hausdorff space. Let $K(X)$ be the collection of all compact subsets of $X$. A topology on $K(X)$ is defined by a subbasis given by sets of the form $I_U=\{K\in K(X)\,|\,K\subset U\}$ and ...
0
votes
0answers
18 views

Show that a bounded set A is relatively compact wrt weak topology iff the closure with the weak*-topology of A in bidual is in X

The problem I am supposed to prove: Let $X$ be Banach. Show that a bounded set $A$ is relatively compact with respect to the weak topology if and only if the closure with the weak$^*$-topology of $A$ ...
-1
votes
0answers
40 views

The Boundary of a Simply Connected Compact Set [duplicate]

I have a simply connected compact set in $\mathbb{R}^2$. I want to know if the boundary of this set is a connected set. I think the answer is yes, but I'm having trouble finding a reference.
-1
votes
1answer
47 views

Boundary of a Connected Compact Set [closed]

I have a connected compact set in $\mathbb{R}^2$. I want to know if the boundary of this set is a connected set.
1
vote
1answer
65 views

$R/Q$ is compact [closed]

$\mathbb R/\mathbb Q$ is known to be compact, where topology on $\mathbb R$ is Euclid topology, and define $a~b$ is equivalent to $a-b\in\mathbb Q$, topology on $\mathbb R/\mathbb Q$ is given by ...
0
votes
0answers
26 views

Uniform continuity and maximum and minimum with a continuous function in $\mathbb{R}^n$.

Problem: Let $f: \mathbb{R}^n \to \mathbb{R}$ a continuous function, so that for every $\epsilon >0$, there exits a compact set $K \subset \mathbb{R}^n$ with $|f(x)|<\epsilon$ for all $x \in \...
0
votes
1answer
30 views

Can we strictly prove that smaller delta will still satisfy epsilon in the proof of continuity?

A video I was watching on the proof that a function continuous over a compact set is necessarily uniformly continuous over that set used the very intuitive idea that if $\left|x-y\right|<\delta_0:\...
-1
votes
2answers
77 views

Prove or disprove that $\overline{A}$ is not compact

I have this exercise where $X$ is a topological Hausdorff space, and we let $A$ be an infinite subset of the topological space. We have that for every $a\in A$ there exists an open subset $U\subseteq ...
1
vote
1answer
64 views

Let X be Hausdorff. If $A\subseteq X$ is compact then $\overline{A}$ is compact

I am doing this exercise where we let X be a topological Hausdorff space. I need to show that if $A\subseteq X$ is compact then the closure $\overline{A}$ is compact as well. Furthermore I have to ...
1
vote
1answer
63 views

Compactness or Connectedness of $A \subseteq \Bbb R^2$

Consider the subsets A and B of $\Bbb R^2$ defined by $A = \{\left(x,x\sin\frac1x\right):x \in(0,1]\}$ and $B=A \cup \{(0,0)\}.$ Then A is compact. A is connected. B is compact. B is connected. ...
-1
votes
1answer
29 views

Behaviour of ${f}^{-1}$ under continuous map

Let $f : \Bbb R \to \Bbb R$ be a continuous function. Which of the following is/are always true ? ${f}^{-1}(A)$ is open for all open sets $A \subseteq \Bbb R$ ${f}^{-1}(A)$ is closed for all closed ...
0
votes
2answers
44 views

If $M$ is compact, then follows from $\forall x \in M | f(x) < a$ that $\sup_{x\in M} f(x)<a$

Let $(X,d)$ be a metric space, $M\subseteq X$ and $f: M \to \mathbb{R}$ continuous. Show that: If $M$ is compact, then follows from $\forall x \in M | f(x) < a$ that $\sup_{x\in M} f(x)<a$. My ...
2
votes
1answer
90 views

An example of a non-Hausdorff space X, and a compact subset, which is not relatively compact

I am trying to find an example of a non-Hausdorff space $X$ and a subset $A$ of $X$, which is compact, but not relatively compact, i.e. the closure $\overline{A}$ of $A$ is not compact. I thought I ...
1
vote
0answers
22 views

Can we cover the area with a finite number of open intervals?

Assume that $A\subset[a,b]$ and there exists $\epsilon>0 , \epsilon<b-a$ such that $l(A)>b-a-\epsilon$, where $l$ is the Lebesgue measure. Let $r>0$ be a real number. Does there then exist ...
0
votes
1answer
24 views

Equicontinuous and compactness

Hello, everyone. I want to discuss about this theorem, which I read in "Principles of Mathematical Analysis" by Rudin, to enrich my knowledge about equicontinuity. I think "...
0
votes
1answer
101 views

How to show $f:Y\to X^{+}$ is continuous? [closed]

I am struggling with an exercise in Topology. The exercise goes: Let $Y$ be Hausdorff, and $X\subseteq Y$ an open subspace which is locally compact. Let $X^{+}$ denote the one-point compactification ...
4
votes
2answers
66 views

Prove that $C+C=[0,2]$, where $C$ is the Cantor set.

I'd like someone to verify my sketch proof of the below exercise 3.3.7 from Abbott's, Understanding Analysis. If it's incorrect, could you hint/point at the correct approach to the proof. Thanks! ...
2
votes
1answer
57 views

Assertions about compact sets and closed sets

I am self-studying Real Analysis from the text, Understanding Analysis by Stephen Abbott. I would like someone to verify my justifications/counterexamples to the following assertions about closed and ...
0
votes
1answer
42 views

Quotient space is compact or not [closed]

I would like to check whether the quotient space is compact or not. I know compact space's quotient space is also compact because projection map is continuous. But, what about the quotient space which ...
4
votes
1answer
71 views

Is the space $\text{Map}(S^1,S^1)$ of continuous maps on $S^1$ compact? (Compact-open topology)

Is the space $\text{Map}(S^1,S^1)$ of continuous maps $S^1\to S^1$ compact? Here $\text{Map}(S^1,S^1)$ has the compact-open topology. I'm not too savvy with the compact-open topology so I'm not sure ...
0
votes
1answer
52 views

Prove that the set $A = \left\{\frac{2n+2}{2n+1} : n \in \mathbb{N}\right\} \cup \left\{0,1\right\}$ is compact

I'm currently studying about metric spaces, specifically on topic about compactness, here's the exercise in my lecture notes In usual metric space $\mathbb R$, Prove that the set $$A = \left\{\frac{...
0
votes
1answer
24 views

Continuity of a supremum function over compact sets

I'm unsure how to formally prove or disprove the following claim. It came up in trying to prove convergence of a gradient descent-style algorithm. I've tried using different definitions of continuity ...
2
votes
1answer
93 views

Open/compact sets in metric $d_1$ and $d_2$

In a set $X$ we consider two metrics $d_1$ and $d_2$. We consider that the identity map $f:(X,d_1)\rightarrow (X,d_2)$ with $f(x)=x$ is continuous. Which of the following statements are correct? (a) ...
1
vote
0answers
21 views

Compactness of set of measures with respect to weak-*topology

I am reading a proof of the Stone-Weierstrass theorem by De Branges, but I am having trouble understanding the following part. Let $E$ be a locally compact Hausdorff space and $C_0(E, \mathbb{C})$ the ...
4
votes
2answers
59 views

Difference between “E is closed set” and “Every infinite subset of E has a limit point in E”

I am reading Principles of mathematical analysis by Walter Rudin and in have come across this theorem: Theorem 2.41: If a set E in $R^k$ has one of the following three properties, then it has the ...
1
vote
1answer
70 views

Is this function locally integrable ? Needs spherical coordinates!

Let $ d\in \mathbb{N}$ be the dimension we consider. Let $\mu$ be a probability density on $\mathbb{R}^d$. Consider vector fields of the form $f:\mathbb{R}^d\to\mathbb{R}^d$ $$ f(x):=\frac{x}{\|x\|^k},...
0
votes
0answers
54 views

Compact space that is not Hausdorff nor locally compact

I'm trying to classify various topological concepts about compactness according to 3 properties: Hausdorff, compactness and locally compactness. Knowing that a Hausdorff and compact space is always ...
4
votes
1answer
58 views

Every compact subset of $\mathbb R^1$ is the support of a Borel measure

I know this already has an answer here, though it is very cryptic. So, I'm making this post for solution/proof-verification (it is not a duplicate) - I've come up with a measure on Borel sets of $\...
0
votes
1answer
35 views

“Expansion” mapping on a compact

Let $(K,d)$ be a compact metric space. We consider $f: K -> K$ : $$\forall x,y \in K: d(x,y) \leq d(f(x), f(y))$$ Show that $$\forall \epsilon >0 :d(f(x), f(y)) \leq d(x, y)+\epsilon$$ What I'...
1
vote
1answer
37 views

Help with proof with One-Point Compactification and Quotient Spaces.

I have been tasked with proving the following: Let $X$ be a compact, Hausdorff space, and let $U$ be a proper opens subset of $X$. Prove that $$ U^{\infty} \cong X / \left( X - U \right)$$ Note that ...
0
votes
1answer
20 views

When is for a continuous function $f:E_1\times E_2\to E_3$, the map $x_1\mapsto f(x_1,\;\cdot\;)$ into $C(E_2,E_3)$ continuous?

Let ($E_i,d_i)$ be a metric space, $\overline d_3:=\min(1,d_3)$ and $$\overline\rho(f,g):=\sup_{x_2\in E_2}\overline d_3(f(x_2),g(x_2))\;\;\;\text{for }f,g\in E_3^{E_2}.$$ Now let $f:E_1\times E_2\to ...
0
votes
0answers
11 views

Proof confirmation on compactness of k-cells.

Looks like this proof works. Thanks for confirming. Theorem : Every k - cell is compact. Proof : Assume the k -cells are wrt the intervals $[a_j,b_j], 1\leq j \leq k$. We prove this by induction. For $...
1
vote
0answers
55 views

$X-\bigcup \lbrace A_{n}: n \in \mathbb{N} \rbrace$ dense in $X$ if $X$ compact Haussdorff and $A_{n}$ nowherdense in $X$ for each $n \in \mathbb{N}$

Let $X$ be a compact Haussdorf topological space . Prove that if $\lbrace A_{n}: n \in \mathbb{N} \rbrace$ is a sequence of nowhere dense sets of $X$. Then $X-\bigcup \lbrace A_{n}: n \in \mathbb{N} \...
-1
votes
1answer
26 views

Is the set compact relative to the following metric space? [closed]

Suppose we have the metric space $(X,d)$ such that $X = \{x\in \mathbb{R}_{++}: x=1/\pi,\pi \in (0,1]\}$, where $\mathbb{R}_{++}$ implies that $x \in X$ is strictly positive. The metric $d$ on $X$ is ...
1
vote
0answers
21 views

Total Boundedness of Finite Sets with 1-Wasserstein Metric

Let $(X,d)$ be a totally bounded metric space and $\mathcal F(X)$ its set of nonempty finite subsets equipped with the following metric: $$ d_W(A, B) = \!\!\! \sup_{\substack{f:X\to\mathbb{R}\\\text{...
0
votes
0answers
46 views

How to prove that open ordinal space (i.e. $[0,\Omega)$) with order topology is countably compact and pseudocompact but not compact?

How to prove that open ordinal space (i.e. $[0,\Omega)$) with order topology is countably compact but not compact, Where $\Omega$ is the first uncountable ordinal? We know that every compact space is ...
0
votes
1answer
34 views

Compact-open topology where domain is discrete or trivial

(1) Let $X$ be a discrete space with $|X|=n$. Show that the space of functions $\text{Map}(X,Y)$ with the compact-open topology is homeomorphic to $Y\times Y\times\cdots\times Y$ (n times). (2) If $X$ ...

1
2 3 4 5
107