Finite product topology

Mathematical statement

In the two-factor exact families qualify each complete finite factor topology, build the full typed Cartesian carrier and every open rectangle U×V, then close under arbitrary unions to obtain the product topology. Each projection has open preimages. A map z↦(f(z),g(z)) from a finite source is continuous exactly when both coordinates are continuous. Finite product and box topologies agree here, but not by an automatic inference for infinite products. A family of basic rectangles is usually not itself the complete topology.

Representative formula (exact TeX)

P(X,Y)=t

Assumptions

• This node requires at least one factor; exact families fix two nonempty factors and assert no zero-factor product convention.
• Both factor carriers are finite, with complete topologies containing empty/full sets and closed under arbitrary unions and finite intersections.
• Ordered pairs, rectangles and families of sets use extensional canonical form.

Symbol scope

• P — finite-product-topology operator
• X — first topological space
• Y — second topological space
• t — product topology

Intuition, definition, and misconceptions

Qualify factor topologies and the Cartesian carrier first.

Close the rectangle basis under arbitrary unions.

Audit projections and coordinate continuity only in the full generated topology.

Basic rectangles are not necessarily all opens, and finite box equality does not extend automatically to infinite products.

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 basis — Generate the topology from open rectangles.
• Topological space — Qualify factor and product axioms.
• Set operations — Use Cartesian products, unions and intersections.
• Binary relations and set projections — Keep the typed ordered-pair carrier and coordinate maps.
• Continuous maps between topological spaces — Audit projections and the coordinate criterion.
• Function composition — Retain coordinate maps and the universal-property background.

Verification boundary

Finite enumeration certifies the specified spaces, not the general product universal property.

not_formalized

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

Next obligation

Add arbitrary products, box topology and Tychonoff boundaries.

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.