This entry is about closed subsets of a topological space. For other notions of “closed space” see for instance closed manifold.
A subspace of a space is closed if the inclusion map is a closed map.
The closure of any subspace is the smallset closed subspace that contains , that is the intersection of all open subspaces of . The closure of is variously denoted , , , , 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 in the locale defines a closed subspace which is given by the closed nucleus
The idea is that this subspace is the part of which does not involve (hence the notation , or any other notation for a complement), and we may identify with when we are looking only away from .
Revised on June 19, 2013 12:08:33
by Urs Schreiber