Basis for a Topology

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A basis (or base) for a topology on a set is a collection of open sets (the basis elements) such that every open set in is the union or finite intersection of members of .

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Equivalently, a collection of open sets is a basis for a topology on if and only if it has the following properties:

1. For each , there is at least one basis element containing .

2. If and , then there is a basis element containing such that .

The set of all open disks contained in an open square form a basis. Drag the point within the square; then drag the centers of the disks and change their radii as needed to illustrate property 2 of a basis.

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Contributed by: Izidor Hafner (July 2016)
Open content licensed under CC BY-NC-SA


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Reference

[1] E. J. Borowski and J. M. Borwein, Collins Dictionary of Mathematics, HarperCollins Publishers, 1989 p. 46.



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