Prove if $C$ and $D$ are non empty disjoint closed subsets of $M$ such that $C$ or $D$ is compact, then $d(C,D)$ > 0
2978
A and B disjoint, A compact, and B closed implies there is positive distance between both sets
55648
Intersection of finite number of compact sets is compact?
22464
Which of the following subsets of   $\mathbb{R}^2$ are compact
136108
Arbitrary intersection of closed, connected subsets of a compact space connected?
14219
Is the intersection of a closed set and a compact set always compact?
102048
Closure of a compact set in Regular space $ X $
52565
Prove that if $d \mid n \in \mathbb{N}$, then $\varphi(d) \mid \varphi(n)$.
147197
Number of connected components of the complement of a compact
98206
Let $B\subset\mathbb{R}^d$ be closed and $z\in\mathbb{R}^d$, show that there is $b_0\in B$ such that $\delta(z,B)=\|z-b_0\|$.
158569
Proving a particular subset of $R^n$ is closed
106957
Compact sets are closed?
70959
Prove that a set $E$ is closed iff it's complement $E^{c}$ is open
65358
Prove $X$ is compact iff for each family $\{ F_\alpha\}_{\alpha\in I}$ of closed subsets of $X$ with the FIP, we have $\cap F_\alpha\neq\varnothing$.
80180
Discrete and compact subset must be finite
82333
let $(X,d)$ be a metric space. How I can show that any finite subset of $X$ is closed.
123149
Prove that $E$ is disconnected iff there exists two open disjoint sets $A$,$B$ in $X$
8015
Compact set plus a closed set is a closed set