A collection shall consists of mathematical objects that may not share some unambiguously properties. This is different from the so-called class, which consists of mathematical objects that share unambiguously properties. However, a class may not form a set. And we call such classes proper classes. What's more, a family is different from a class. A indexed family or simply family is a collection of mathematical objects that is indexed by a set called index set.

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This paragraph was originally published at June 20, 2021 when I was an undergraduate student.