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relevance to linear algebra (vector spaces)
''Basis is a [[mathematics]] term.''
''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
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.
#if x is in the intersection of 2 basis elements ''B<sub>1</sub>'' and ''B<sub>2</sub>'', then it is in some basis element ''B<sub>3</sub>'', where ''B<sub>3</sub>'' is a subset of ''B<sub>1</sub> ∩ B<sub>2</sub>''.
#if x is in the intersection of 2 basis elements ''B<sub>1</sub>'' and ''B<sub>2</sub>'', then it is in some basis element ''B<sub>3</sub>'', where ''B<sub>3</sub>'' is a subset of ''B<sub>1</sub> ∩ B<sub>2</sub>''.
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''.
[[category:topology]]
When the topological space is a [[vector|vector space]] V, the basis generates V under the operations of [[addition]] and [[scalar]] multiplication.
[[category:topology]][[category:algebra]]