§ 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".