focal point



topology (point-set topology, point-free topology)

see also algebraic topology, functional analysis and topological homotopy theory


Basic concepts

Universal constructions

Extra stuff, structure, properties


Basic statements


Analysis Theorems

topological homotopy theory



A point of a topological space is called focal (Freyd-Scedrov) if its only open neighbourhood is the entire space.


  • The closed point \bot of Sierpinski space 2\mathbf{2} is a focal point.

  • The vertex vv of a Sierpinski cone (or scone) s(X)s(X) on a space XX, given by a pushout in TopTop

    1×X 1 ×1 X v 2×X s(X),\array{ 1 \times X & \to & 1 \\ \mathllap{\bot \times 1_X} \downarrow & & \downarrow \mathrlap{v} \\ \mathbf{2} \times X & \to & s(X), }

    is a focal point. This construction is in fact the same as generically adding a focal point to XX.

  • The prime spectrum of a ring AA has a focal point iff AA is a local ring. In this case, the focal point is given by the unique maximal ideal of AA.


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 XX is called local if in any covering of XX by opens U iU_i, at least one U iU_i is XX.

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

𝒪(X)Ω,U{|U=X}, \mathcal{O}(X) \to \Omega, \quad U \mapsto \{ \star | U = X \},

where Ω\Omega is the frame of opens of the point.


  • Peter Freyd, A. Scedrov, Geometric logic, (North-Holland, Amsterdam)

Revised on December 30, 2013 11:38:28 by Ingo Blechschmidt (