Definition:Almost All/Set Theory

Definition

Countable

Let $S$ be a countably infinite set.

Let $P: S \to \set {\text {true}, \text {false} }$ be a property of $S$ such that:

$\set {s \in S: \neg \map P s}$

is finite.

Then $P$ holds for almost all of the elements of $S$.

Uncountable

Let $S$ be an uncountably infinite set.

Let $P: S \to \set {\text {true}, \text {false} }$ be a property of $S$ such that:

$\set {s \in S: \neg \map P s}$

is countable (either finite or countably infinite).

Then $P$ holds for almost all of the elements of $S$.

Also defined as

Some sources do not distinguish between $S$ being a countably infinite set and an uncountably infinite set, and merely state that $P$ holds for almost all elements of $S$ if $\set {s \in S: \neg \map P s}$ is finite.

Sources

• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 5$: Limits