Fort space

In mathematics, there are a few topological spaces named after M. K. Fort, Jr.

Fort space¹ 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:

• A does not contain p, or
• A contains all but a finite number of points of X.

The subspace ${\displaystyle X\setminus \{p\}}$ has the discrete topology and is open and dense in X. The space X is homeomorphic to the one-point compactification of an infinite discrete space.

Modified Fort space² 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:

• A contains neither p nor q, or
• A contains all but a finite number of points of X.

The space X is compact and T₁, but not Hausdorff.

Fortissimo space³ 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:

• A does not contain p, or
• A contains all but a countable number of points of X.

The subspace ${\displaystyle X\setminus \{p\}}$ 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 Lindelöfication⁴ of an uncountable discrete space.