nLab
open subspace

Contents

Definition

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

For a point-based notion of space such as a topological space, an open subspace is the same thing as an open subset.

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

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

The idea is that this subspace is the part of XX which involves only UU, and we may identify VV with UVU \Rightarrow V when we are looking only at UU.

The interior of any subspace AA is the largest open subspace contained in AA, that is the union of all open subspaces of AA. The interior of AA is variously denoted Int(A)Int(A), Int X(A)Int_X(A), A A^\circ, A\overset{\circ}A, etc.

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

Revised on May 2, 2012 16:45:08 by Urs Schreiber (82.113.99.15)