A point of a topological space is called focal (Freyd-Scedrov) if its only open neighbourhood is the entire space.
The closed point of Sierpinski space is a focal point.
The vertex of a Sierpinski cone (or scone) on a space , given by a pushout in
is a focal point. This construction is in fact the same as generically adding a focal point to .
The category of sheaves over (the site of open subsets) of a topological space with focal point is a local topos.
Every topos has a free “completion” to a “focal space”, given by its Freyd cover.
- Peter Freyd, A. Scedrov, Geometric logic, (North-Holland, Amsterdam,
Revised on December 7, 2011 19:27:44
by Urs Schreiber