Basis

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Basis is a mathematics term.

Linear algebra[edit]

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[edit]

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

1. each x in X, is in at least one basis element.
2. if x is in the intersection of 2 basis elements B₁ and B₂, then it is in some basis element B₃, where B₃ is a subset of B₁ ∩ B₂.

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.