topology_axioms

std.topology.topology_axioms · Level L3

The axioms of a topology on a finite set, given as the list of its open sets τ (an m × n 0/1 matrix), as 1 or 0: the empty set and the whole set are open, and the union and the intersection of any two open sets are open. For a finite family, pairs suffice for every union and every finite intersection; the empty union is ∅ and the empty intersection is X. Each test runs over all pairs, by set sizes: the smallest open set containing U and V must have |U ∪ V| points, and the largest inside both |U ∩ V|.

∅, X ∈ τ; U, V ∈ τ ⇒ U ∪ V, U ∩ V ∈ τ

Signature

topology_axioms(tau: f64[m, n]) → f64[], f64[], f64[], f64[]

Structure

The function as NOVA stores it: one box per input, operation and output, and arrows that carry values. A double border marks another library function this one runs — called once, or by Scan once per element; select it to open that function.

input
operation
constant
call
output

Verification

Signature proven by NOVA’s shape solver, for every size.
Equal to the reference the rows as Python frozensets: every union and intersection of a pair in exact rational arithmetic, on all 40 test cases.
All 160 float64 results are exact: the error is zero.
Interpreter and NumPy backend return bit-identical results.
At float32 too: the 40 test cases rounded to float32, and 40 of them exact, with zero error.

Accuracy in detail

correctly rounded (the float64 nearest the exact value): 100%
bit-equal to the NumPy formula in float64: 100%
largest error, in units in the last place: 0

Identity

Calls

—

Called by

sigma_algebra_axioms is_topology

sha256:b6edb39e3e48081124d6e57b47904140bea4f93d8875d2facefb6d98793b054d

The semantic hash of the graph. It changes when the program changes, and never when only its documentation does.

Control handle

Symbol: Ω:std.topology.topology_axioms · Ω:topology_axioms
Pins: sha256:030457202805be0ff1e2a36424249e8cffb1df6f194516e94db1c4a47048afe7this graph alone
Evidence: sha256:e68653be48b3fb87dfd4a069fbf155db6cc8fd89b9754f6a55122154291dcc6fthe hash of its verification record
Needs: no capability: a pure function

Through NOVA’s control layer, the symbol launches this function only while the program still matches what it pins: a change to this graph, or to any graph it reaches, needs a migration first.