nLab
basis

There are different notions in mathematics called basis. Generally speaking, these fall into two classes:

• situations where the term base is also used:

  • topological base
  • basis for a Grothendieck topology,
  • etc …;

• the generating set of a free object:

  • basis of a free module,
  • basis of a vector space
  • basis in functional analysis (which is applied more generally than to free objects, but analogously)
  • etc ….

Note that the plurals of ‘basis’ and ‘base’ are both spelt ‘bases’ (although their pronunciations differ).