Definition 8

A family $(A_i)_{i \in I}$ of subsets of a topological space $X$ is said to be locally finite if for each $x \in X$ there is a neighbourhood $V$ of $x$ such that $V \cap A_i = \varnothing$ for all but a finite number of indices $i \in I$. A set $\mathscr{S}$ of subsets of $X$ is said to be locally finite if the family of subsets defined by the identity map of $\mathscr{S}$ onto itself is locally finite.

It is clear that if $(A_i)_{i \in I}$ is a locally finite family of subsets and if $B_i \subset A_i$ for each $i \in I$, then the family $(B_i)_{i \in I}$ is locally finite.

Every finite family of subsets of a topological space $X$ is obviously locally finite; the converse is not true in general.

\* For example, in $\mathbf{R}$, the open covering formed by the interval ]\leftarrow, 1[ and the intervals ]$n, \rightarrow[$ for each integer $n \geqslant 0$ is locally finite; and each interval ]$n, \rightarrow[$ meets an infinite number of sets of this covering. \*