Basis for a Topology
Basis for a Topology
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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B
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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:
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1. For each , there is at least one basis element containing .
x∈X
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x
2. If ,∈B and , then there is a basis element ∈B containing such that ⊂⋂.
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2
x∈⋂
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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.