# 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.

5,331 questions
Filter by
Sorted by
Tagged with
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 ...
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 ...
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 ...
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 ...
1answer
25 views

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 ...
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 $H$,...
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 ...
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 ...
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 $A$ ...
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.
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.
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 ...
0answers
26 views

2answers
77 views

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'...
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 ...
1answer
20 views

0answers
55 views