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  • $\begingroup$ In set theory, the cardinal number $2^n$ is defined to be the cardinality of the set of functions from a set of $n$ elements to a set of $2$ elements. Writing the latter set as $\{0,1\}$ (with no loss of generality) shows these functions can be identified with the indicator functions on $\mathcal B$--but an indicator function is determined by, and determines, the subset of $\mathcal B$ on which it equals $1,$ QED. In this way the quoted statement makes sense even when $\mathcal B$ is countably infinite. $\endgroup$
    – whuber ♦
    Commented Nov 18, 2019 at 17:27