Let $X$ be a Hausdorff space. Then for all $x \in X$, the set $\{x\}$ is closed. Moreover, every finite subset of $X$ is closed.
138039
If $X$ is Hausdorff then every finite set 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.
50846
Is a set without limit points necessarily closed?
148213
Given a topological space $X$, why are both $X$ and $\emptyset$ open and closed?
111141
Why is every discrete subgroup of a Hausdorff group closed?
11405
If a nonempty set of real numbers is open and closed, is it $\mathbb{R}$? Why/Why not?
70049
Irreducible topological spaces and irreducible Hausdorff space
111360
Topological spaces in which every proper closed subset is compact
94825
Open set as a finite union of closed sets
22854
Let $A , B\subseteq\mathbb{R}$. If $A$ is closed and $B$ is compact, is $A\cdot B$ closed?
100560
$X$ compact Hausdorff space, characterize the maximal ideals of $C(X)$
65180
A compact Hausdorff space that is not metrizable
66334
How is every subset of the set of reals with the finite complement topology compact?
59275
If $X$ is an infinite set, then $X$ with the discrete topology is not compact
106957
Compact sets are closed?
144383
The closure of a connected set in a topological space is connected
100476
Why does a Hausdorff but not countably compact space have an infinite closed discrete subset?