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 prime spectrum of a ring has a focal point iff is a local ring. In this case, the focal point is given by the unique maximal ideal of .
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.
In locale theory
A locale is called local if in any covering of by opens , at least one is .
The locale associated to an sober space is local in this sense if and only if the space possesses a focal point (see Johnstone, discussion preceding lemma C1.5.6). Locale theoretically, this point is then given by the frame homomorphism
where is the frame of opens of the point.
- Peter Freyd, A. Scedrov, Geometric logic, (North-Holland, Amsterdam)