is_basis

std.topology.is_basis · Level L3

Whether a finite family B of subsets is a basis for some topology, as 1 or 0: its members cover X, and for any two members, each point of their intersection lies in a member inside the intersection. A Scan of basis_step over the members. A basis is not a subbasis: the subbasis condition asks only that the members cover.

∪B = X; x ∈ B₁ ∩ B₂ ⇒ ∃B₃ ∈ B: x ∈ B₃ ⊆ B₁ ∩ B₂

Signature

is_basis(B: f64[m, n]) → 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 cover, and the intersection condition for every pair and point in exact rational arithmetic, on all 40 test cases.
All 40 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

basis_step

Called by

is_basis_for

sha256:6bceee1922c9fb7fcc8a8f0b22a6e670d0967e1a358e04e5306cef2a995067d0

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.is_basis · Ω:is_basis
Pins: sha256:98f79c51ef83f9780bf995422c406e83b92033903ac488d16a2a908bdfd56370this graph and the 1 it reaches through calls
Evidence: sha256:79a472a6f4979929d94ccc55eb24cc75a0e001cec7465d5f8583d176e6a6a7dathe 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.