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A finite subcover is a subcover with finitely many members, including possibly none when $K$ is the empty set. Requiring a finite subcover for every open cover is the definition of compactness.

A subcover of an open cover $\{U_i:i\in I\}$ of a subset $K$ is a selection of its members whose union still contains $K$. The selected open sets retain their original sizes; selecting a subcover does not mean replacing them by smaller open sets.