Makkai workship Birthday offering
  Billed to the organisers as “a short talk ... offering a peculiar topological space that Peter Johnstone and I used some 30 years ago to solve a topoidal problem.”

The space? — Heath’s V-space, a retopologised real upper half-plane.

The double cover? — Concocted by emulating one construction of the connected double cover of the circle.

The motivation? — To display a topos (that of locally finite — i.e., decidable and Kuratowski-finite — sheaves over Heath’s V-space) where subobjects of 1 generate and internal choice holds ((SG) & (IC)) yet supports do not split ((SS) fails, and with it, (AC)). [>>]