# Every open set in a four-point product [← Guide index](https://ocme.evemisslab.com/en/derivations/) · [Original MKO](https://ocme.evemisslab.com/en/mko/mko-finite-product-topology/) This is an owner-adopted natural-language teaching guide. Finite/arithmetic checks are not universal proofs; no Lean was run, and review scope is recorded per revision. Original source status labels are retained as historical data. Formulas retain exact TeX; no new semantic AST is invented. derivation-mko-finite-product-topology-01 · ocme-new142-reader-adoption-20261021 ## Goal and scope Generate every product open from two Sierpinski factors and audit projections. Two nonempty finite factors; no zero-factor convention or infinite-box extrapolation. ## Givens and symbols - sets - topology - definition_here - G1 — X=Y={0,1}, both topologies are {empty,{0},{0,1}}; a=(0,0),b=(0,1),c=(1,0),d=(1,1), P={a,b,c,d}. - P — Cartesian carrier - \mathcal B — rectangle basis - \pi_i — coordinate projection ## Rules and sources - mko_definition · cited — 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. · mko-finite-product-topology - sets · prerequisite — Compute unions, intersections, complements, inclusion and preimages element by element. - topology · prerequisite — A topology contains empty and whole sets and is closed under arbitrary unions and finite intersections; a finite family admits exhaustive checks. - definition_here · prerequisite — The Cartesian carrier consists of ordered pairs; product opens are arbitrary unions of open rectangles; a projection selects its designated coordinate; continuity requires every target-open preimage to be open. ## Main derivation S01 · impliesBefore \varnothing\subset\{0\}\subset\{0,1\} After \tau_X,\tau_Y\text{ are topologies} Conditions: none Rule: topology · Uses: G1 Leaf basis: topology Each three-member chain contains endpoints and is union/intersection closed, qualifying both factors. S02 · definitionBefore P=X\times Y After P=\{a,b,c,d\},\quad |P|=2\cdot2=4 Conditions: none Rule: definition_here · Uses: G1 Leaf basis: definition_here The complete Cartesian carrier contains four ordered pairs. S03 · definitionBefore U\times V,\quad U\in\tau_X,V\in\tau_Y After \mathcal B=\{\varnothing,\{a\},\{a,b\},\{a,c\},P\} Conditions: none Rule: definition_here · Uses: G1, S01, S02 Leaf basis: definition_here Deduplicate nine rectangles into five sets: an empty factor gives empty, and the four nonempty combinations are listed. S04 · definitionBefore \tau_P=\{\bigcup\mathcal C:\mathcal C\subseteq\mathcal B\} After \tau_P=\{\varnothing,\{a\},\{a,b\},\{a,c\},\{a,b,c\},P\} Conditions: none Rule: definition_here · Uses: S03 Leaf basis: definition_here The union of the two incomparable rectangles adds {a,b,c}; every other union is already listed, giving exactly six opens. S05 · impliesBefore \{a,b\}\cap\{a,c\}=\{a\},\quad \{a,b\}\cup\{a,c\}=\{a,b,c\} After \tau_P\text{ is a topology} Conditions: none Rule: topology · Uses: S04 Leaf basis: topology Except for the incomparable pair, inclusion determines unions/intersections; operations with the added three-point set also stay listed, and endpoints are present. S06 · impliesBefore \pi_1^{-1}(\{0\})=\{a,b\},\quad\pi_2^{-1}(\{0\})=\{a,c\} After \pi_1,\pi_2\text{ are continuous} Conditions: none Rule: definition_here · Uses: G1, S04, definition_here Leaf basis: definition_here The remaining factor opens are empty and whole, with preimages empty and P; every required preimage is open. S07 · impliesBefore h(z)=(f(z),g(z)),\quad f,g\text{ continuous} After h^{-1}(U\times V)=f^{-1}(U)\cap g^{-1}(V)\text{ open} Conditions: f:Z\to X,\ g:Z\to Y\text{ continuous} Rule: definition_here · Uses: S03, S04, definition_here Leaf basis: definition_here For a finite topological source Z, continuous coordinates give open preimage intersections; preimages commute with arbitrary rectangle unions. S08 · impliesBefore h\text{ continuous} After f=\pi_1\circ h,\quad g=\pi_2\circ h\text{ continuous} Conditions: h:Z\to P\text{ continuous} Rule: definition_here · Uses: S06, S07, definition_here Leaf basis: definition_here Composition preserves continuity because preimages compose; together with the previous step this gives the coordinate criterion. ## Substeps and lemmas No substeps in this guide. ## Reverse audits - S08 · premise_audit · S06, S07, definition_here — The criterion uses the complete six-open product topology, not the unclosed rectangle list. - S07 · premise_audit · S03, S04, definition_here — The topology of Z and continuity of both coordinates are premises of this conditional argument. - S06 · premise_audit · G1, S04, definition_here — All three target opens have been checked for each projection. - S05 · premise_audit · S04 — Every arbitrary union in this finite family reduces to repeated binary unions. - S04 · premise_audit · S03 — The added member is not a rectangle, protecting against omitted unions. - S03 · premise_audit · G1, S01, S02 — The rectangle family is not yet called the complete topology. - S02 · premise_audit · G1 — (0,1) and (1,0) remain distinct. - S01 · premise_audit · G1 — Both coordinate types of the ordered pairs are fixed. ## Error controls and negative controls Union closureBuild rectangles and then all their unions, not just the basis. Types and directionProjection preimages use the corresponding coordinate's opens. - The five rectangles are all opens. → reject — {a,b,c} is a union of two rectangles and is not itself a rectangle. - Cartesian pairs are unordered. → reject — b and c are different points. - Finite product=box implies equality for infinite products. → reject — In the countably infinite product of R indexed by the natural numbers (the real-sequence space), the box-open product with every factor (-1,1) is not product-open: any basic product neighborhood of zero restricts only finitely many coordinates, so an unrestricted coordinate can be changed to 2. ## Result and limitations The product has exactly six opens and continuous projections; paired maps from finite sources satisfy the coordinate criterion. ## Verification and source binding 1 declared checks for this guide; only the listed expressions are covered. - source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f - adopted source SHA-256: b45b64af72af44b80a818bcda104e7319061c6713cd7fd49a7869539b4038479 - original source SHA-256: b45b64af72af44b80a818bcda104e7319061c6713cd7fd49a7869539b4038479 - Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW [Same-source full JSON (formulas, ASTs, controls and revision log)](https://ocme.evemisslab.com/data/derivations/coverage-20261021/guides/derivation-mko-finite-product-topology-01.json)