In the topos of locally finite sheaves over Heath’s V-space,
then, (SS) fails (and with it, (AC)), yet (IC) holds
(because every object is Kuratowski-finite and decidable), as does (SG)
(because Heath’s V-space has a base of clopen sets).
Thus this topos refutes the reversibility of the first and third of the implications
implications each of which is reversible for Grothendieck topoi.
(For [necessarily] non-Grothendieck topoi refuting the
reversibility of the remaining implications here, and for more
implications among other conjunctions of
similar topos-theoretic conditions, to whose converses
the present topos of locally finite sheaves over Heath’s V-space provides a counterexample, see the 30-year-old work
mentioned at the outset.)
F.E.J. Linton • Wesleyan Univ. Math/CS Emeritus