When moving from metric spaces to general topological spaces, these objects must be generalized to nets. The Borel-Cantelli lemma follows by constructing one of these objects containing events and then applying continuity of the measure. In general, compactness in the sense of a finite subcover for every open cover is independent of a type of compactness named for these objects. The (*) Cantor set is homeomorphic to the space of these objects over the discrete space {0, 1} (“zero comma one”), and Cantor’s diagonalization argument builds one of these objects out of countably many of them where the new one differs from each original in at least one place. If a is one of these objects, one might say “a sub n approaches x as n approaches infinity”. For 10 points, a collection of objects indexed by the natural numbers is referred to by what term that titles an example named for Fibonacci? ■END■
ANSWER: sequences [accept Fibonacci sequence; accept sequentially compact space; prompt on descriptions like function from the natural numbers or infinite lists before mentioned]
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