Basis (topology)

A basis for a topology is a collection of open sets such that every open set is a union of basis members. It provides a compact way to specify all open sets on a space.

Basis criterion: A collection of subsets is a basis when it covers every point and each pairwise intersection can be locally refined by a member. These two conditions let basis members generate a topology.
Open interval: An interval of real numbers that excludes both endpoints. Open intervals form the standard basis for the real line.
Second-countable space: A topological space with a countable basis. Countable bases make many topological arguments manageable.
Closed set: A subset whose complement is open in a given topological space. Bases generate open sets directly; closed sets follow by taking complements.
Topology generated by a basis: The topology consisting of all unions of members of a given basis. It is the topology produced by the collection in the definition.
Euclidean space: A real coordinate space equipped with its usual distance and topology. Open balls in Euclidean space provide a canonical basis.
First-countable space: A topological space in which every point has a countable local base. Local bases determine when sequences can capture neighborhood behavior.
Neighborhood: A set containing an open set that contains a specified point. Basis members refine neighborhoods but need not themselves be every neighborhood.
Open set: A subset designated as open by a topology, with openness governed by axioms such as closure under unions and finite intersections. Every open set must be expressible as a union of basis members.
Open ball: The set of points whose distance from a fixed center is less than a fixed radius. Open balls generate the usual topology on metric spaces.

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