Subbasis for a topological space

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This article is about a basic definition in topology.
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Definition

A subbasis for a topological space is a collection of open subsets of the topological space such that the following equivalent conditions hold:

  • The set of subsets obtained by taking finite (possibly empty) intersections of these, form a basis for the topological space
  • Every open subset is a countable union of finite intersections of these
  • The topology on the space is the coarsest topology for which the given subsets are all open

No constraints are there for a collection of subsets to form a subbasis.

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