# Definition:Fort Space

## Definition

Let $S$ be an infinite set.

Let $p \in S$ be a particular point of $S$.

Let $\tau_p \subseteq \mathcal P \left({S}\right)$ be a subset of the power set of $S$ defined as:

$\tau_p = \left\{{U \subseteq S: p \in \complement_S \left({U}\right)}\right\} \cup \left\{{U \subseteq S: \complement_S \left({U}\right)}\right.$ is finite$\left.{}\right\}$

That is, $\tau_p$ is the set of all subsets of $S$ whose complement in $S$ either contains $p$ or is finite.

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

### Countable Fort Space

Let $S$ be countably infinite.

Then $\tau_p$ is a countable Fort topology, and $\left({S, \tau_p}\right)$ is a countable Fort space.

### Uncountable Fort Space

Let $S$ be uncountable.

Then $\tau_p$ is an uncountable Fort topology, and $\left({S, \tau_p}\right)$ is an uncountable Fort space.

## Also see

• Results about Fort spaces can be found here.

## Source of Name

This entry was named for Marion Kirkland Fort, Jr.