Definition:Modified Fort Space

Definition
Let $N$ be an infinite set.

Let $\left\{{a}\right\}$ and $\left\{{b}\right\}$ be singleton sets such that $a \ne b$ and $a, b \notin N$.

Let $S = N \cup \left\{{a}\right\} \cup \left\{{b}\right\}$.

Let $\tau_{a, b}$ be the set of subsets of $S$ defined as:
 * $\tau_{a, b} = \left\{{H \subseteq N}\right\} \cup \left\{{H \subseteq S: \left({a \in H \lor b \in H}\right) \land N \setminus H}\right.$ is finite$\left.{}\right\}$

That is, a subset $H$ of $S$ is in $\tau_{a, b}$ either:
 * $(1): \quad H$ is any subset of $N$

or:
 * $(2): \quad$ if $a$ or $b$ or both are in $H$, then $H$ is in $S$ only if it is cofinite in $S$, that is, that it contains all but a finite number of points of $S$ (or $N$, equivalently).

Then $\tau_{a, b}$ is a modified Fort topology on $a$ and $b$, and the topological space $T = \left({S, \tau_{a, b}}\right)$ is a modified Fort space.

Also see

 * Modified Fort Topology is Topology