Definition:Fort Space

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Let $S$ be an infinite set.

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

Let $\tau_p \subseteq \powerset S$ be a subset of the power set of $S$ defined as:

$\tau_p = \leftset {U \subseteq S: p \in \relcomp S U} \text { or } \set {U \subseteq S: \relcomp S U}$ is finite$\rightset{}$

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 $T = \struct {S, \tau_p}$ is a Fort space.

Countable Fort Space

Let $S$ be countably infinite.

Then $\tau_p$ is a countable Fort topology, and $\struct {S, \tau_p}$ is a countable Fort space.

Uncountable Fort Space

Let $S$ be uncountable.

Then $\tau_p$ is an uncountable Fort topology, and $\struct {S, \tau_p}$ 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.