Let $(X, d)$ be a metric space. Let $A, B ⊆ X$ be nonempty disjoint subsets and suppose $A$ is compact and $B$ is closed.
2978
A and B disjoint, A compact, and B closed implies there is positive distance between both sets
147213
Prove that finite unions of compact sets are compact
17067
To prove that $(\mathbb{P(N)},d)$ is a metric space
11114
Is a closed subset of a compact set (which is a subset of a metric space $M$) compact?
152057
Let $(X,d)$ be a compact metric space. Let $f: X \to X$ be such that $d(f(x),f(y)) = d(x,y)$ for all $x,y \in X$. Show that $f $ is onto (surjective).
154978
Some equivalent formulations of compactness of a metric space
128894
About compactness and completeness of set $X = \mathbb{R}$ with the metric $d(x, y) = \frac{|x-y|}{1+|x-y|}$
42698
Set of limit points of a subset of a Hausdorff space is closed.
64694
$C_{c}(X)$ is complete. then implies that $X$ is compact.
7402
Showing a compact metric space has a countable dense subset
99645
Metrizable topological space $X$ with every admissible metric complete then $X$ is compact
36237
Closed set mapped to itself in a compact Hausdorff space
102168
Prove that there exists disjoint open subsets $U,V$ of $X$ such that $x \in U$ and $B \subseteq V$.
17842
Is every set in a separable metric space the union of a perfect set and a set that is at most countable?
28804
No. of possible dense subsets of a metric space
14219
Is the intersection of a closed set and a compact set always compact?
114608
Proof concerning regular space: there exist a closed set contained in any open set containing $x$