Subbasis for a topological space

This article is about a basic definition in topology.
VIEW: Definitions built on this | Facts about this | Survey articles about this
View a complete list of basic definitions in topology

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.

Related notions

• Basis for a topological space