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 $\perp$ of Sierpinski space $2$ is a focal point.

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

  $$\begin{array}{ccc}1\times X& \to & 1\\ \perp \times 1_X\downarrow & & \downarrow v\\ 2\times X& \to & s(X),\end{array}$$\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 $X$.

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.

References

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