§ 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 , consider the cone: (ie, take product with and smash all the together).
- Given any topological space , 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".