# Basis

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**Basis**as edited by FredericBernard (Talk | contribs) at 15:57, September 11, 2017. This URL is a permanent link to this version of this page.*Basis is a mathematics term.*

## Linear algebra

In linear algebra, a basis is a set of linearly independent vectors that spans a vector space *V*. Any vector in the vector space can then be written as a linear combination of the basis.

This usage of the term is similar to its common usage: a basis is the foundation for what is needed.

## Topology

A **basis** *B* for a topology *T* on a set *X* is a collection of subsets of *X* (called **basis elements**) such that

- each
*x*in*X*, is in at least one basis element. - if x is in the intersection of 2 basis elements
*B*and_{1}*B*, then it is in some basis element_{2}*B*, where_{3}*B*is a subset of_{3}*B*._{1}∩ B_{2}

If *B* satisfy the above 2 conditions, then the **topology T generated by B** is the collection of subsets

*U*of

*X*such that for each

*x*in

*U*, there is a basis element

*V*in

*B*such that

*x*is in

*V*and

*V*is a subset of

*U*.