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) “Internal Choice” holds (i.e., epimorphisms split locally).
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 those in (AC) ⇒ (B) & (SS) ⇒ (B) & (SG) , so counterexamples must be non-Grothendieck.]
  The topos: locally finite sheaves over Heath’s V-space.
(Local finiteness ⇒ (IC); 0-dimensionality of the base ⇒ (SG).)
  Same topos serves as counterexample to the reversibility of several other implications among similar conjunctions of topos-theoretic conditions. [For details, consult the 30-year-old work mentioned at the outset.]