basis

noun

1. 1

   A physical base or foundation.

2. 2

   A starting point, base or foundation for an argument or hypothesis.

3. 3

   An underlying condition or circumstance.

4. 4

   A regular frequency.

   “Cars must be checked on a yearly basis.”

5. 5

   In a vector space, a linearly independent set of vectors spanning the whole vector space.

6. 6

   Amount paid for an investment, including commissions and other expenses.

7. 7

   A collection of subsets ("basis elements") of a set, such that this collection covers the set, and for any two basis elements which both contain an element of the set, there is a third basis element contained in the intersection of the first two, which also contains that element.

   “The collection of all possible unions of basis elements of a basis is said to be the topology generated by that basis.”