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What is Base (topology)?

In mathematics, a base (or basis) B of a topology on a set X is a collection of subsets of X such that every finite intersection of elements of B (including X itself, which is, by a standard convention, the empty intersection) is a union of elements of B. A base defines (one says also generates) a topology on X that has, as open sets, all unions of elements of B. Bases have been introduced because some topologies have a base consisting of open sets that have specific useful properties. This is typically the case for the Zariski topology on the spectrum of a ring. For the usual basis of this topology, every finite intersection of basis elements is a basis element. Therefore bases are sometimes required to be stable by finite intersection.

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 — Date merged: 11/6/2021, 1:32:50 PM
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