# 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. ${\ displaystyle F}$${\ displaystyle G \ subseteq F}$

## 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 ${\ displaystyle V}$ ${\ displaystyle K}$${\ displaystyle U}$${\ displaystyle V}$${\ displaystyle V}$${\ displaystyle V}$

• ${\ displaystyle U \ neq \ emptyset}$
• for everyone too (closeness to the addition) and${\ displaystyle u, v \ in U}$${\ displaystyle u + v \ in U}$
• for everyone and everyone too (seclusion regarding the scalar multiplication )${\ displaystyle \ alpha \ in K}$${\ displaystyle u \ in U}$${\ displaystyle \ alpha \ cdot u \ in U}$

applies.

### Topological space

${\ displaystyle (X, {\ mathcal {O}})}$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 . ${\ displaystyle X}$${\ displaystyle {\ mathcal {O}}}$${\ displaystyle U \ subseteq X}$${\ displaystyle U}$${\ displaystyle X}$${\ displaystyle \ left (U, \ left \ {O \ cap U \, | \, O \ in {\ mathcal {O}} \ right \} \ right)}$

This subspace generally does not inherit all properties of the larger space , for example the separation property T 4 can be lost. ${\ displaystyle (X, {\ mathcal {O}})}$

### Metric space

${\ displaystyle \ left (X, d \ right)}$be a metric space . Each subset becomes a subspace by restricting the metric from to . ${\ displaystyle U \ subseteq X}$${\ displaystyle (U, d | _ {U \ times U})}$${\ displaystyle X \ times X}$${\ displaystyle U \ times U}$

If is a complete metric space, then is a complete metric space if and only if is closed. ${\ displaystyle \ left (X, d \ right)}$${\ displaystyle (U, d | _ {U \ times U})}$${\ displaystyle U \ subseteq X}$

## 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.