Totally-imperfect space

A topological space not containing a subset homeomorphic to a Cantor set. For instance, any complete separable uncountable metrizable space contains an uncountable subspace that, as well as its complement, is totally imperfect.

Comments

A subset of $\mathbf R$ such that it and its complement are totally imperfect is usually called a Bernstein set, after F. Bernstein [a1]. Such sets are non-measurable.

References

[a1]	F. Bernstein, "Zur Theorie der trigonometrischen Reihe" Ber. K. Sächs. Ges. Wissenschaft. Leipzig Math.-Phys. Kl. , 60 (1908) pp. 325–338