A construction of containing spaces

In this paper we give a set-theoretical construction of containing spaces for an arbitrary collection of spaces. This construction gives a satisfactory, in our opinion, answer to Problem 7 given in Section 3 of Arhangelʹskiı̆ and Fedorchuk (Encyclopaedia of Math. Sci., Vol. 17, 1990, p. 1). Using this construction we also introduce the notion of a saturated class of spaces and prove that the class of all T0-spaces, the class of all regular spaces, and the class of all completely regular spaces are saturated classes.