Difference between revisions of "Basis"
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m (New page: 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. ...) |
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#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>''. | ||
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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''. | ||
[[category:topology]] | [[category:topology]] | ||
Revision as of 05:07, April 12, 2007
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.