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.

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