Transfinite dimensions Ind_m

摘要

We investigate dimensions Ind_m (m is an integer ≥ 2 or m = ∞) introduced in [V.V. Fedorchuk, Weakly infinite-dimensional spaces, Uspekhi Mat. Nauk 62 (2) (2007) 109-164]. These dimensions have intrinsic properties similar to those of the classical transfinite dimension Ind = Ind_2. In particular, (1) Ind_m X < ω_1 for every countable dimensional metrizable compactum X; (2) every normal space X has a compactification bX with ωbX = ωX and Ind_m bX ≤ Ind_m X. Moreover, if Ind X is defined (respectively Ind X is finite), then Ind_m X is defined (respectively Ind_m X is finite) for every m.