derivation-mko-finite-product-topology-01 · ocme-new142-reader-adoption-20261021
Generate every product open from two Sierpinski factors and audit projections.
Two nonempty finite factors; no zero-factor convention or infinite-box extrapolation.
setstopologydefinition_hereG1 — 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 projectionmko_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-topologysets · 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.Before
\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.
Before
P=X\times Y
After
P=\{a,b,c,d\},\quad |P|=2\cdot2=4Conditions: none
Rule: definition_here · Uses: G1
Leaf basis: definition_here
The complete Cartesian carrier contains four ordered pairs.
Before
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.
Before
\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.
Before
\{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.
Before
\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.
Before
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.
Before
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.
No substeps in this guide.
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.Build rectangles and then all their unions, not just the basis.
Projection preimages use the corresponding coordinate's opens.
reject — {a,b,c} is a union of two rectangles and is not itself a rectangle.reject — b and c are different points.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.The product has exactly six opens and continuous projections; paired maps from finite sources satisfy the coordinate criterion.
1 declared checks for this guide; only the listed expressions are covered.
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