$S$ is a compact subset of $\mathbb{R}$ and $T$ is a closed subset of $S$ $\implies$ $T$ compact
11114
Is a closed subset of a compact set (which is a subset of a metric space $M$) compact?
47810
$C(X)$ is separable when $X$ is compact?
122663
Proving $T$ and $U$ are equal if I know $(X,T)$ and $(X,U)$ are hausdorff and compact
125962
The set $A'$ is closed in a $T_1$ space
3708
How can I prove that compact subset iff sequentially compact in second countable topological spaces?
46970
If $X$ is compact and $T:X \to Y$ continuous and bijective, show that $T$ is homeomorphism.
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?
28637
Proving continuous image of compact sets are compact
140570
Suppose $E\subseteq\mathbb{R}$ is closed and $K\subseteq\mathbb{R}$ is compact. Show that every open cover of $E\cap K$ has a finite subcover
65358
Prove $X$ is compact iff for each family $\{ F_\alpha\}_{\alpha\in I}$ of closed subsets of $X$ with the FIP, we have $\cap F_\alpha\neq\varnothing$.
29303
Cartesian product of compact sets is compact
18596
A set $A \subset l_1$ is compact if and only if closed, bounded, and one other condition
59194
Spaces in which "$A \cap K$ is closed for all compact $K$" implies "$A$ is closed."
58948
If $A$ is compact, is then $f(A)$ compact?
43376
If $A$ is compact and $B$ is closed, show $d(A,B)$ is achieved
110744
How to prove that a compact set in a Hausdorff topological space is closed?