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).

Definitions and examples

Vector subspaces

These are very well behaved; as a vector space XX is simply a module over a field, so a subspace of XX is simply a submodule. More generally, this is a special case of a subalgebra.

Vector subspaces are precisely the subobjects in Vect.

Topological subspaces

Given a topological space XX (in the sense of Bourbaki, that is: a set XX and a topology τ X\tau_X) and a subset YY of XX, a topology τ Y\tau_Y on a set YY is said to be the topology induced by the set inclusion YXY\subset X if τ Y=τ X pw{Y}={UYUτ X}\tau_Y = \tau_X \cap_{pw} \{Y\} = \{ U \cap Y | U\in\tau_X \}. The pair (Y,τ Y)(Y,\tau_Y) is then said to be a (topological) subspace of (X,τ X)(X,\tau_X).

If a continuous map f:ZXf:Z\to X is a homeomorphism onto its image f(Z)f(Z) in the induced topology on f(Z)f(Z), this inclusion map is sometimes called an embedding; ZZ is thus isomorphic in Top to a subspace of XX.

See at topological subspace.

Topological vector subspaces

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 LL, which can also be thought of as a frame, a sublocale of LL is given by a nucleus on the frame LL. Even if LL is topological, so that LL can be identified with a sober topological space, still there are generally many more sublocales of LL than the topological ones.




For Grothendieck topologies, one instead of a subspace has a concept of a subsite?.

Revised on February 28, 2014 14:20:48 by Urs Schreiber (