On 𝑄 sets

Fleissner, William G.; Miller, Arnold W.

doi:10.1090/S0002-9939-1980-0550513-4

On $Q$ sets
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by William G. Fleissner and Arnold W. Miller PDF

Proc. Amer. Math. Soc. 78 (1980), 280-284 Request permission

Abstract:

A Q set is an uncountable set X of the real line such that every subset of X is an ${F_\sigma }$ relative to X. It is known that the existence of a Q set is independent of and consistent with the usual axioms of set theory. We show that one cannot prove, using the usual axioms of set theory: 1. If X is a Q set then any set of reals of cardinality less than the cardinality of X is a Q set. 2. The union of a Q set and a countable set is a Q set.

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On separability and metrizability of spaces with Souslin’s condition

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Current research on Q sets