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Definition: Index Set and Set Family
Let $X$ be an arbitrary set, and let $I$ be a nonempty set, called the index set. A sets family $X_{i}{\text{ , }}i\in I$ is a map $I\to \mathcal P(X),$ where $\mathcal P(X)$ is the power set of $X.$
Notes
 In other words, each $X_i$ is a subset of $X.$
 The elements $i\in I$ are called indices of the family of sets $X_{i}{\text{ , }}i\in I.$
Mentioned in:
Corollaries: 1
Definitions: 2 3 4 5 6 7 8 9 10 11 12 13 14
Proofs: 15 16 17 18
Propositions: 19
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References
Adapted from CC BYSA 3.0 Sources:
 Brenner, Prof. Dr. rer. nat., Holger: Various courses at the University of Osnabrück