nLab
focal point

Context

Topology

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

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

Definition

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

Examples

  • 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.

Properties

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.

References

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

Revised on December 30, 2013 11:38:28 by Ingo Blechschmidt (46.244.180.181)