Topological basis

Mathematical statement

Every basis element is a subset of X and their union covers X. If x lies in B1∩B2, some B3 contains x and lies inside that intersection. A basis for an existing topology additionally consists of open sets and generates every open set as a union. A basis must not be confused with a subbasis.

Representative formula (exact TeX)

B(X)

Assumptions

• The complete finite carrier and full candidate basis family are supplied.
• Existing-topology exercises list every open set and check all topology axioms exactly.

Symbol scope

• B — topological basis
• X — carrier

Intuition, definition, and misconceptions

Coverage and local intersection refinement are both required.

The generated topology consists of all unions of basis elements, including the empty union.

An existing topology needs both openness of basis elements and generation of every open set.

An arbitrary cover is not automatically a basis, and a subbasis cannot simply be treated as a basis.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Topological space — A topology is generated or compared.
• Open and closed sets — Basis elements and open sets must be related correctly.
• Set operations — Coverage, intersections and unions are separate constructions.

Verification boundary

Exact enumeration certifies only the declared finite-carrier instance.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Add subspace, product and quotient topologies.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.