finite subcover
Given a cover with possibly infinitely many patches, a finite subcover is a small selection — only finitely many of the original patches — that still manages to cover everything. It is the act of saying “most of these umbrellas are redundant; these few already do the job.”
Formally, if {U_i} (i in some index set I) is an open cover of X, a finite subcover is a finite subset {U_i1, ..., U_in} of the same family whose union still contains X. Crucially the patches must be chosen from the given cover — you may not invent new open sets, only keep finitely many of the ones already offered.
Whether a finite subcover can always be extracted is the dividing line between compact and non-compact spaces. Compactness demands that every open cover, no matter how cunningly arranged, admit a finite subcover. The strength of compactness comes from this universal quantifier: it is not enough that some cover trims down; all of them must.
Cover [0, 1] by the open sets (-1, 0.7) and (0.3, 2). Already this two-set subfamily covers [0, 1], so it is a finite subcover. Compactness of [0, 1] says such a finite subcover exists no matter what open cover you start from — even a cover with infinitely many tiny patches.
Two open sets already cover [0,1]; compactness guarantees a finite subcover from every cover.