Of interest to topos theorists ... because it shows that supports need not split (“(SS) fails”) even in a topos where
(SG) Subobjects of 1 Generate, and
(IC) locally, epimorphisms split (“Internal” axiom of Choice).
In particular, epimorphisms need not actually split, i.e., (AC) may fail, i.e., the implication (AC) => (IC) & (SG) is not reversible.
[NB: in contrast, for Grothendieck topoi, each of the implications (AC) => (IC) & (SG) => (B) & (SG) is reversible, as are (AC) => (B) & (SS) => (B) & (SG) , so any counterexample is necessarily non-Grothendieck.]
  The topos: locally finite sheaves over Heath’s V-space.
(Local finiteness => (IC); 0-dimensionality of the base => (SG).)
  Same topos can then be used as counterexample to the reversibility of another couple of handfuls of implications among other conjunctions of similar topos-theoretic conditions.
[For details, consult that 30-year-old reference mentioned at the outset.]

[ Ref. ]