This entry is about closed subsets of a topological space. For other notions of “closed space” see for instance closed manifold.
CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
A subspace $A$ of a space $X$ is closed if the inclusion map $A \hookrightarrow X$ is a closed map.
The closure of any subspace $A$ is the smallset closed subspace that contains $A$, that is the intersection of all open subspaces of $A$. The closure of $A$ is variously denoted $Cl(A)$, $Cl_X(A)$, $\bar{A}$, $\overline{A}$, etc.
(There is a lot more to say, about convergence spaces, smooth spaces, schemes, etc.)
For a point-based notion of space such as a topological space, a closed subspace is the same thing as a closed subset.
A subset is closed precisely if
In locale theory, every open $U$ in the locale defines a closed subspace which is given by the closed nucleus
The idea is that this subspace is the part of $X$ which does not involve $U$ (hence the notation $U'$, or any other notation for a complement), and we may identify $V$ with $U \cup V$ when we are looking only away from $U$.