A Universe of Sorts

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".

§ Focal Point