Difference between revisions of "Basis"

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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''.
 
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''.
 
When the topological space is a [[vector|vector space]] V, the basis generates V under the [[continuous]] operations of [[addition]] and [[scalar]] multiplication. The [[dimension]] of V is given by the number of elements in the basis, up to intersection and multiplicity.
 
  
 
[[category:topology]][[category:linear algebra]]
 
[[category:topology]][[category:linear algebra]]

Revision as of 02:34, September 1, 2008

Basis is a mathematics term.

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.