On transfinite inductive dimension and deficiency modulo a class P

We establish some fundamental properties of transfinite inductive dimension modulo a class P and use them to compute the dimensions of Bernstein sets and other pathological spaces. We show that transfinite completeness degree and transfinite completeness deficiency do not agree on separable metrizable spaces. We construct a separable compact space that contains an ∞-Inductionally embedded discrete subspace, which answers a question of Aarts and Nishiura (1993).