Various more or less geometrical concepts are called spaces, to name a few vector spaces, topological spaces, algebraic spaces, …. If such objects form a category, it is natural to look for the subobjects and to call them subspaces. However, often the natural subspaces in the field are the regular subobjects; convsersely, it is also often the case that variants which are not subobjects in categorical sense are allowed, such as an immersed submanifold? (whose image topological subspace is not a manifold in general).
Given a topological space (in the sense of Bourbaki, that is: a set and a topology ) and a subset of , a topology on a set is said to be the topology induced by the set inclusion if . The pair is then said to be a (topological) subspace of .
See at topological subspace.
A ‘subspace’ of a topological vector space usually means simply a linear subspace, that is a subspace of the underlying discrete vector space.
However, the subspaces that we really want in categories such as Ban are the closed linear subspaces. (Essentially, this is because we want our subspaces to be complete whenever our objects are complete.)
Given a locale , which can also be thought of as a frame, a sublocale of is given by a nucleus on the frame . Even if is topological, so that can be identified with a sober topological space, still there are generally many more sublocales of than the topological ones.
… submanifold …
For Grothendieck topologies, one instead of a subspace has a concept of a subsite?.