Geodesic
On the Cardinality of the Family of Precomplete Classes in~$P_E$
Matematičeskie zametki, Tome 78 (2005) no. 6, pp. 864-869

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Let $E$ be an infinite set of cardinality $\mathbf m$, and let $P_E$ be the set of all functions defined on $E$. We prove that the cardinality of the family of all classes precomplete in $P_E$ is equal to $2^{2^{\mathbf m}}$. If $C_{\mathbb R}$ is the set of all continuous functions of real variables, then the cardinality of the family of all classes precomplete in $C_{\mathbb R}$ is equal to $2^{2^{\aleph_0}}$. 
@article{MZM_2005_78_6_a5,
     author = {S. S. Marchenkov},
     title = {On the {Cardinality} of the {Family} of {Precomplete} {Classes} in~$P_E$},
     journal = {Matemati\v{c}eskie zametki},
     pages = {864--869},
     publisher = {mathdoc},
     volume = {78},
     number = {6},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2005_78_6_a5/}
}
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S. S. Marchenkov. On the Cardinality of the Family of Precomplete Classes in~$P_E$. Matematičeskie zametki, Tome 78 (2005) no. 6, pp. 864-869. http://geodesic.mathdoc.fr/item/MZM_2005_78_6_a5/