Definition:Modified Fort Space

Definition

Let $N$ be an infinite set.

Let $\set a$ and $\set b$ be singleton sets such that $a \ne b$ and $a, b \notin N$.

Let $S = N \cup \set a \cup \set b$.

Let $\tau_{a, b}$ be the set of subsets of $S$ defined as:

$\tau_{a, b} = \set {H \subseteq N} \cup \set {H \subseteq S: \paren {a \in H \lor b \in H} \land N \setminus H \text { is finite} }$

That is, a subset $H$ of $S$ is in $\tau_{a, b}$ if and only if 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 $T = \struct {S, \tau_{a, b} }$ is a modified Fort space.

Also see

• Modified Fort Topology is Topology

• Definition:Fort Space

• Results about modified Fort spaces can be found here.

Source of Name

This entry was named for Marion Kirkland Fort, Jr.

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $27$. Modified Fort Space