Any countable topological vector space over a finite field admits a closed discrete basis.
Engelking,General Topology, 2nd ed
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Any infinite Abelian topological group of prime exponent has an infinite maximally almost periodic subgroup.
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Any countable topological $\mathbb F_p$-vector space has a closed discrete basis
Any countable topological vector space over a finite field admits a closed discrete basis.
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Maximally almost periodic subgroups of Abelian groups of prime exponent
Any infinite Abelian topological group of prime exponent has an infinite maximally almost periodic subgroup.