# Tagged Questions

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.

36 views

### A topological space is countably compact iff every countably infinite subset has a limit point

A topological space is countably compact iff every countably infinite subset has a limit point. I'm completely stuck on this one. The book is recommending to use the fact that a space is countably ...
14 views

### All small enough subsets of a compact metric spaces are covered by one element of its open cover [duplicate]

$(X,d)$ is a compact metric space, $\{U_i | i \in I\}$ is an open cover of $X$. I need to show that there is a number $\delta > 0$, such that if $A \subset X$ and diameter of $A \le \delta$, then ...
27 views

52 views

30 views

### The standard n-simplex is compact set

$" Let \ K\subseteq \mathbb R^n \ be \ a \ compact \ set,\ then\ the\ convex \ hull \ of\ K\ is\ also\ compact \ set\ "$ In order to prove this we use that the standard n-simpex as defined ...
32 views

### is $[0,1]\backslash \mathbb{Q}$ totally bounded?

I learnt totally bounded by myself. Now, I am still trying to understand the definition and looking for counterexamples which is totally bounded but not compact. The below is some of counterexamples: ...
43 views

43 views

25 views

### "Correct'' morphism extension to Nagata compactifications

Can a morphism of separated schemes of finite type over a field be extended to Nagata compactifications of the schemes preserving the closed complements? Let $\mathbf{Sch}/k$ be the category of ...
29 views

25 views

### Show that for all compact $K$ and for all continuous function $f:X \to K$, there is $g: \overline{e(X)} \to K$ continuous with $g \circ e = f$.

Let $X$ a Tychonoff space, $S =${$f:X \to [0,1] : f$ continuous} and consider the topology immersion $e: X \to \prod_{s \in S} [0,1]$ where $e(x) = (f(x))_{f \in S}, \quad \forall x \in X$. Show ...
48 views

16 views

### Issue with compactness implies boundedness proof

The proof is outlined as follows: (copied from Wikipedia but Apostol gives the same idea) If a set is compact, then it is bounded: Consider the open balls centered upon a common point, with any ...
I've been studying for my final exam in a general topology course, and I came upon this problem about compactness that I'm have a really tough time solving. Let $a$ and $b$ be integers, with $b\neq 0$...
If $E$ is a normed space and $F$ is a subspace of $E$, how to prove that if $F\neq\{0\}$ then $F$ is not compact? I begin by this let $x\in F$ then $F=\bigcup_{x\in F} B(x,\varepsilon)$ how to say ...