Basis

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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

  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.

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