Basis and Subbasis: Concepts, Formulas and Examples

A basis is a small collection of open sets from which every open set is built as a union, like building blocks for a topology. A subbasis is even smaller: take finite intersections of its members first, then unions. Instead of listing uncountably many open sets, you just describe the generators.

This matters because topologies are usually too large to write down. The usual topology on R has uncountably many opens, yet the basis of open intervals describes it completely. Checking continuity or compactness reduces to checking basis or subbasis elements, turning impossible verifications into routine ones.

Key Concepts

• Basis: opens such that every open set is a union of them; every point of every open set lies in a basis element inside it.
• Basis criterion: covers X, and the intersection of two members is a union of members.
• Generated topology: all unions of basis members; always a topology when the criterion holds.
• Subbasis: any collection S; its topology is unions of finite intersections of members of S.
• Product topology: R^2 gets the basis of open rectangles (a,b) × (c,d).

Formulas

• Topology from a basis: tau = {unions of subfamilies of B}.
• Topology from a subbasis: finite intersections first, then unions.
• Continuity via subbasis: f is continuous iff preimages of subbasis elements are open: far less to check.

Worked Example

Open intervals are a basis for the usual topology on R; the rays (-∞, a), (b, ∞) are a subbasis for the same topology.

1. Coverage: x ∈ (x-1, x+1) for every real x.
2. Intersections: (a,b) ∩ (c,d) = (max(a,c), min(b,d)), an interval or empty. Criterion holds.
3. Every usually-open U is a union of intervals: each x ∈ U has an interval around x inside U.
4. Subbasis: (-∞, b) ∩ (a, ∞) = (a, b); unions of these give every interval, then every open set.
5. Two families of rays thus replace all intervals: subbases are economical.

Common Misconceptions

• Not every collection is a basis: {(0,2), (1,3)} alone does not cover R, failing the criterion.
• Basis elements must be open in the topology they generate, by construction.
• A subbasis need not be closed under intersections; that is the point: intersections are manufactured from it.

Real-World Applications

• Function spaces: infinite products get topologies from subbases of coordinate constraints.
• Probability: weak topologies from subbases of measurement maps give the right convergence.
• Digital images: pixel neighbourhoods act as basis elements for digital topologies.

A basis generates a topology by unions, a subbasis by finite intersections then unions. Open intervals are a basis for R's usual topology, while the rays (-∞, a), (b, ∞) do the same job as a subbasis. Generators make topology computable in practice.