A subspace of a space is open if the inclusion map 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 in the locale defines an open subspace which is given by the open nucleus
The idea is that this subspace is the part of which involves only , and we may identify with when we are looking only at .
The interior of any subspace is the largest open subspace contained in , that is the union of all open subspaces of . The interior of is variously denoted , , , , etc.
(There is a lot more to say, about convergence spaces, smooth spaces, schemes, etc.)