nLab focal point

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 $\mathbf{2}$ is a focal point.

• The vertex $v$ of a Sierpinski cone (or scone) $s(X)$ on a space $X$, given by a pushout in $Top$

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

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

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

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 $X$ is called local if in any covering of $X$ by opens $U_i$, at least one $U_i$ is $X$.

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

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

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

References

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

• Peter Johnstone, Sketches of an Elephant