nLab subspace

Idea

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; conversely, it is also often the case that variants which are not subobjects in the categorical sense are allowed, such as an immersed submanifold?. (These may have self-intersections, and then the immersion map is not monic, nor can this map be replaced by the inclusion of the image, since this image is usually not a manifold.)

Definitions and examples

Vector subspaces

These are very well behaved; as a vector space $X$ is simply a module over a field, so a subspace of $X$ 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 $X$ (in the sense of Bourbaki, that is: a set $X$ and a topology $\tau_X$) and a subset $Y$ of $X$, a topology $\tau_Y$ on a set $Y$ is said to be the topology induced by the set inclusion $Y\subset X$ if $\tau_Y = \tau_X \cap_{pw} \{Y\} = \{ U \cap Y | U\in\tau_X \}$. The pair $(Y,\tau_Y)$ is then said to be a (topological) subspace of $(X,\tau_X)$.

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

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.)

Sublocales

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

Submanifolds

… submanifold …

Subsites

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

Related concepts

• subobject

• linear subspace