Difference between revisions of "Basis"
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| + | ''Basis is a [[mathematical]] term.'' | ||
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A '''basis''' ''B'' for a topology ''T'' on a set ''X'' is a collection of subsets of ''X'' (called '''basis elements''') such that | A '''basis''' ''B'' for a topology ''T'' on a set ''X'' is a collection of subsets of ''X'' (called '''basis elements''') such that | ||
#each ''x'' in ''X'', is in at least one basis element. | #each ''x'' in ''X'', is in at least one basis element. | ||
Revision as of 18:18, April 18, 2007
Basis is a mathematical term.
A basis B for a topology T on a set X is a collection of subsets of X (called basis elements) such that
- each x in X, is in at least one basis element.
- 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.