Binary Interior Excision and Nonuniqueness of Graph Homology

We give an explicit nonuniqueness example for the \((J_1,\times)\) Eilenberg–Steenrod axioms on Čech closure spaces, where \(J_1\) is the indiscrete two-point interval and \(\times\) is the categorical product. Applying the known clique and closed neighborhood complex constructions to the mutual singleton graph gives two integral relative homology theories, with subspace complexes formed intrinsically. For every binary interior cover, both constructions preserve the relevant union and intersection, so excision is an actual relative chain isomorphism.

The theories share point coefficient group \(\mathbb Z\), homotopy invariance, the long exact sequence of a pair and arbitrary coproduct additivity, but on the undirected four-cycle they give the homology of \(S^1\) and \(S^2\). Their restrictions to finite graph pairs also respect ordinary interval homotopies. The complexes and general closure-space nonuniqueness are established prior work; the focus is the explicit relative binary-cover argument and its graph consequence. This AI-assisted, self-audited, unrefereed note does not settle every strengthened axiom variant, and makes no absolute priority claim.