Topological Basis

A topological basis is a subset B of a set T in which all other open sets can be written as unions or finite intersections of B. For the real numbers, the set of all open intervals is a basis.

Stated another way, if X is a set, a basis for a topology on X is a collection B of subsets of X (called basis elements) satisfying the following properties.

1. For each x in X, there is at least one basis element B containing x.

2. If x belongs to the intersection of two basis elements B_1 and B_2, then there is a basis element B_3 containing x such that B_3 subset B_1 intersection B_2.

(Munkres 2000).

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Munkres, J. R. Topology: A First Course, 2nd ed. Upper Saddle River, NJ: Prentice-Hall, 2000.

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Topological Basis

Cite this as:

Weisstein, Eric W. "Topological Basis." From MathWorld--A Wolfram Web Resource.

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