# Subspace

In mathematics, space is a set with a mathematical structure . A **subspace** or **subspace** is understood to be a subset which is closed in terms of structure in the broadest sense. The exact definition depends on the structure.

## Examples

### Subspace

Let be a vector space over a body . A subset of is called a subspace of if it is itself a vector space with the connections induced by . This is the case if and only if

- for everyone too (closeness to the addition) and
- for everyone and everyone too (seclusion regarding the scalar multiplication )

applies.

### Topological space

be a topological space on the set with the family of open sets . Each subset becomes a subspace if the averages of with the in open sets are defined as open sets of the subspace. thus becomes a topological space that carries the subspace topology .

This subspace generally does not inherit all properties of the larger space , for example the separation property T _{4 can be} lost.
_{}

### Metric space

be a metric space . Each subset becomes a subspace by restricting the metric from to .

If is a complete metric space, then is a complete metric space if and only if is closed.

## Categorical definition

In the context of a category of spaces, a subspace of a space is defined by the fact that a certain monomorphism exists in the space in which it is supposed to be contained. Depending on the situation, one demands, for example, that the monomorphism must be extreme . This makes a difference in non-balanced categories , for example in the category of topological spaces: Every continuous injection is a monomorphism there, but this is not necessarily an embedding in the sense of topology, since the image of a monomorphism can also be coarser than the potential Subspace. An extreme monomorphism, on the other hand, is a topological embedding.

## literature

- Boto von Querenburg:
*Set theoretical topology.*Springer-Verlag, ISBN 3-540-67790-9 - Gerd Fischer:
*Lineare Algebra*, Vieweg-Verlag, ISBN 3-528-03217-0