Subspaces
# Subspaces

## 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?.