Compact dispersed spaces and the $\alpha$-left property

Abstract:

The concept of an $\alpha$-left space was introduced by Arhangel’skij in [A$_{1}$], where, among other results, it was shown that every ${T_2}$ compact $\alpha$-left space is dispersed. In [A$_{2}$] an example was given to show that, assuming the continuum hypothesis CH, not every ${T_2}$ compact dispersed space is $\alpha$-left. The aim of this article is to obtain some sufficient conditions for ${T_2}$ space to be $\alpha$-left and to construct a large class (which contains all products of uncountable ordinals) of compact ${T_2}$ dispersed spaces which are not $\alpha$-left.