Fort space

In mathematics, there are a few topological spaces named after M. K. Fort, Jr..
Fort space.
Fort space is defined by taking an infinite set "X", with a particular point "p" in "X", and declaring open the subsets "A" of "X" such that:
Note that the subspace formula_1 has the discrete topology and is open and dense in "X".
"X" is homeomorphic to the one-point compactification of a infinite discrete space.
Modified Fort space.
Modified Fort space is similar but has two particular points. So take an infinite set "X" with two distinct points "p" and "q", and declare open the subsets "A" of "X" such that:
The space "X" is compact and T1, but not Hausdorff.
Fortissimo space.
Fortissimo space is defined by taking an uncountable set "X", with a particular point "p" in "X", and declaring open the subsets "A" of "X" such that:
Note that the subspace formula_1 has the discrete topology and is open and dense in "X". The space "X" is not compact, but it is a Lindelöf space. It is obtained by taking an uncountable discrete space, adding one point and defining a topology such that the resulting space is Lindelöf and contains the original space as a dense subspace. Similarly to Fort space being the one-point compactification of an infinite discrete space, one can describe Fortissimo space as the "one-point Lindelöfication" of an uncountable discrete space.