Any open or closed subset of a locally compact space is also locally compact
128730
Open subspaces of locally compact Hausdorff spaces are locally compact
22464
Which of the following subsets of   $\mathbb{R}^2$ are compact
144383
The closure of a connected set in a topological space is connected
154390
Stone-Čech compactification. A completely regular topological space is locally compact iff it is open in its Stone-Čech compactification.
26563
Help completing a proof regarding an infinite family of nested closed subsets in a compact metric space
158498
$A\subset \mathbb{R}$ with more than one element and $A/ \{a\}$ is compact for a fixed $a\in A$
106957
Compact sets are closed?
94825
Open set as a finite union of closed sets
36237
Closed set mapped to itself in a compact Hausdorff space
73154
Compact subset in colimit of spaces
16654
Unions and Intersections of Open Sets are Open
11383
Disjoint compact sets in a Hausdorff space can be separated
106534
"Closed subsets of compact space are compact" with an alternative definition of covering
114608
Proof concerning regular space: there exist a closed set contained in any open set containing $x$
111360
Topological spaces in which every proper closed subset is compact
11114
Is a closed subset of a compact set (which is a subset of a metric space $M$) compact?
136108
Arbitrary intersection of closed, connected subsets of a compact space connected?