quotient_open

std.topology.quotient_open · Level L3

Whether a subset U of Y is open in the quotient topology of a surjection q : X → Y, as 1 or 0: its preimage q⁻¹(U) is open in X. Only saturated open sets of X are preimages, so taking images of open sets would in general be wrong.

U open ⇔ q⁻¹(U) ∈ τ_X

Signature

quotient_open(tauX: f64[m, n], q: i64[n], U: f64[k]) → 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 preimage among the open sets of X 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

is_open

Called by

—

sha256:92a563bcde39e80b139b106cba3423f203d1fe25d6110302741494169258b591

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.quotient_open · Ω:quotient_open
Pins: sha256:a43ecf050995af0acdf9d91396730ea47ba988cc7acbceac886fe77cdb57b370this graph and the 1 it reaches through calls
Evidence: sha256:ddb85a4d1f3930aa02da05696c6aebadbc38c29ac5bba5b6213c5b930f1f79d1the 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.