## § Focal point

- The focal point of a space is a point whose only open nbhd is the whole space.
- In the sierpiski space
`(), bottom`

, the `bottom`

is the focal point. - In a local ring, the focal point is given by the maximal ideal (in the prime spectrum, ofc).
- Given any topological space $T$, consider the cone: (ie, take product with $[0, 1]$ and smash all the $\{0\} \times *$ together).
- Given any topological space $T$, now build the scone: take the product with the sierpinski space, and smash everything with the closed point. Then, the apex of the cone / the closed point becomes a focal point for the topological space. This can be seen as a "one point focalization".