Locally finite family

Remarks

Example: in $\R$, the family $([n, n + 1])_{n \in \Z}$ is locally finite. For $x \in \R$, the neighbourhood $(x - 1, x + 1)$ meets $[n, n + 1]$ only when $x - 2 < n < x + 1$, which holds for at most three integers $n$.

Non-example: in $\R$, the family $((0, 1/n))_{n \ge 1}$ is not locally finite. Every neighbourhood of $0$ contains an interval $(-\varepsilon, \varepsilon)$, which meets every $(0, 1/n)$.

If $(A_i)_{i \in I}$ is locally finite, then the closure of $\bigcup_i A_i$ equals $\bigcup_i \overline{A_i}$. In particular, the union of a locally finite family of closed sets is closed.