nLab
closed subspace

This entry is about closed subsets of a topological space. For other notions of “closed space” see for instance closed manifold.


Contents

Definition

A subspace AA of a space XX is closed if the inclusion map AXA \hookrightarrow X is a closed map.

The closure of any subspace AA is the smallset closed subspace that contains AA, that is the intersection of all open subspaces of AA. The closure of AA is variously denoted Cl(A)Cl(A), Cl X(A)Cl_X(A), A¯\bar{A}, A¯\overline{A}, etc.

(There is a lot more to say, about convergence spaces, smooth spaces, schemes, etc.)

Topological spaces

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

Locales

In locale theory, every open UU in the locale defines a closed subspace which is given by the closed nucleus

j U:VUV. j_{U'}\colon V \mapsto U \cup V .

The idea is that this subspace is the part of XX which does not involve UU (hence the notation UU', or any other notation for a complement), and we may identify VV with UVU \cup V when we are looking only away from UU.

Revised on June 19, 2013 12:08:33 by Urs Schreiber (82.169.65.155)