A set $\{S_i \mid i \in I\}$ of sets is said to cover another set $S$ if $S \subset \bigcup_{i \in I} S_i$.

The notion of covering is extremely broad, and mathematicians are often interested in covers where particular restrictions are placed on the $S_i$. For example, if we have only finitely many of the $S_i$ (the index set $I$ is finite), we have a finite cover. If $I$ is countable, we have a countable cover. In topology, one may be interested in the case that the $S_i$ are open sets, in which case we have an open cover.

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