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 subsets of topological spaces.

Others define point finite only for covers.

In each case, however, the underlying meaning is identical.

Also see

• Definition:Metacompact Space
• Definition:Locally Finite Cover

Sources

• 1955: John L. Kelley: General Topology: Chapter $5$: Problem $\text{V}$
• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $3$: Compactness: Paracompactness
• 2000: James R. Munkres: Topology (2nd ed.): Exercise $39.2$