# 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.

5,745 questions
Filter by
Sorted by
Tagged with
41 views

### If $X \subset \mathbb{R}^n$ is compact then $X \times X \subset \mathbb{R}^n \times \mathbb{R}^n$ is compact - proof verification

Claim: If $X \subset \mathbb{R}^n$ is compact, then $X \times X \subset \mathbb{R}^n \times \mathbb{R}^n$ is compact. Key definitions: A set $X$ is closed if every convergence sequence converges to ...
1 vote
37 views

### How to show that the Alexander Subbase Theorem is ZF-equivalent to the Compactness Theorem for first order logic?

Alexander Subbase Theorem (ASB): Let $X$ be a topological space. $X$ is compact if and only if there is a subbase $\mathcal{B}$ for the topology of $X$ such that every subcollection of $\mathcal{B}$ ...
76 views

1 vote
44 views

55 views

36 views

### Compact Riemann surface is sequentially compact.

Now, I try to prove that; M:a compact Riemann surface. $\forall \{P_j\}_{j\in N}\subset M$ (sequence of points) $\exists\{P_{j_k}\} _{k\in N}$ (subsequence of $\{P_j\}$) s.t. the subsequence converge....
1 vote
### Minimum distance from a point to a closed set in $\mathbb{R}^n$.
The task: S is a non-empty closed subset of $\mathbb{R}^n$ equipped with the Euclidean metric. Take $a \not \in S$. Show $\min \{ d(x,a) \ | \ x \in S \}$ exists. Below is my attempt. I wanted to ...