How do I prove that a closed subset of a compact set is compact directly from the definition?
11114
Is a closed subset of a compact set (which is a subset of a metric space $M$) compact?
106534
"Closed subsets of compact space are compact" with an alternative definition of covering
149220
If every compact set is closed, then is the space Hausdorff?
130975
Given a pair of continuous functions from a topological space to an ordered set, how to prove that this set is closed?
73154
Compact subset in colimit of spaces
29303
Cartesian product of compact sets is compact
125962
The set $A'$ is closed in a $T_1$ space
19856
Prove that a subset of a separable set is itself separable
135157
Totally bounded, complete $\implies$ compact
28637
Proving continuous image of compact sets are compact
91991
How do I prove that two closed sets are separable?
47810
$C(X)$ is separable when $X$ is compact?
111538
If $S \subseteq X$ is closed, is $f(S,r)$ necessarily closed?
39293
Given a compact set $K$ and closed set $E,$ prove: $E\cap K\neq \emptyset $
481
Can compact sets completey determine a topology?
100476
Why does a Hausdorff but not countably compact space have an infinite closed discrete subset?
22854
Let $A , B\subseteq\mathbb{R}$. If $A$ is closed and $B$ is compact, is $A\cdot B$ closed?