If $(X,d)$ is a metric space, prove that intersection of any collection of compact sets in $(X,d)$ is compact.
74311
Proving that the Intersection of any Collection of Compact Sets is Compact
125580
Proving that every countable metric space is disconnected?
126944
Metric spaces problem
42266
Show that $d(x,y)=\min \{1,|x-y|\}$ is a metric on $\mathbb R$
41408
Which of the following sets are compact:
26215
Proof Verification: Let $D=\{ \frac{1}{n} \mid n\in\mathbb{N} \} \cup\{0\} \subset \mathbb{E}^1$. Prove that $D$ is compact.
74657
Can this intuition give a proof that an isometry $f:X \to X$ is surjective for compact metric space $X$?
8015
Compact set plus a closed set is a closed set
49374
Closure of a subset in a metric space
73154
Compact subset in colimit of spaces
26563
Help completing a proof regarding an infinite family of nested closed subsets in a compact metric space
28804
No. of possible dense subsets of a metric space
59194
Spaces in which "$A \cap K$ is closed for all compact $K$" implies "$A$ is closed."
133152
(X,d) be a compact metric space. For every open cover, show there exists ε > 0 such that ∀𝑥 ∈ X, B(x,ε) is contained in some member of the cover.
11383
Disjoint compact sets in a Hausdorff space can be separated
52014
Showing $(C[0,1], d_1)$ is not a complete metric space
148403
Compact space and Hausdorff space