X and Y are compact metric spaces. Show that X $\times$ Y is compact
29303
Cartesian product of compact sets is compact
84004
Prove there is no contraction mapping from compact metric space onto itself
74657
Can this intuition give a proof that an isometry $f:X \to X$ is surjective for compact metric space $X$?
102048
Closure of a compact set in Regular space $ X $
14509
Contraction Map on Compact Normed Space has a Fixed Point
148994
$K \subset \mathbb{R}^n$ is compact iff it is closed and bounded
69777
Show that, for every metric $d$, the metrics $d/(1+d)$ and $\min\{1,d\}$ are equivalent
26563
Help completing a proof regarding an infinite family of nested closed subsets in a compact metric space
17067
To prove that $(\mathbb{P(N)},d)$ is a metric space
51206
If $X$ and $Y$ are sets, their cartesian product $X\times Y$ is a set.
74311
Proving that the Intersection of any Collection of Compact Sets is Compact
15480
Open set in a metric space is union of closed sets
65358
Prove $X$ is compact iff for each family $\{ F_\alpha\}_{\alpha\in I}$ of closed subsets of $X$ with the FIP, we have $\cap F_\alpha\neq\varnothing$.
47810
$C(X)$ is separable when $X$ is compact?
28637
Proving continuous image of compact sets are compact
123039
If $0<\alpha < 1$ and $d(x,y)$ is metric, show that $((d(x,y))^{\alpha}$ is metric.
7402
Showing a compact metric space has a countable dense subset