Definition:Point Finite

Definition

Let $S$ be a set.

Let $\CC$ be a set of subsets of $S$.

Then $\CC$ is point finite if and only if each element of $S$ is an element of finitely many sets in $\CC$:

$\forall s \in S: \card {\set {C \in \CC: s \in C} } < \infty$

Also defined as

Some sources define point finite only for a cover of a given set $S$, rather than for a general set of subsets of $S$.

Also see

• Definition:Point Finite Cover

• Results about point finiteness can be found here.

Sources

• 1955: John L. Kelley: General Topology: Chapter $5$: Problem $\text{V}$
• 2000: James R. Munkres: Topology (2nd ed.): Exercise $39.2$