On the base dimension I and the property of universality

Abstract

In Iliadis (2005) [13] for an ordinal α the notion of the so-called ( b n - Ind ⩽ α ) -dimensional normal base C for the closed subsets of a space X was introduced. This notion is defined similarly to the classical large inductive dimension Ind. In this case we shall write here I ( X , C ) ⩽ α and say that the base dimension I of the space X by the normal base C is less than or equal to α. The classical large inductive dimension Ind of a normal space X, the large inductive dimension Ind 0 of a Tychonoff space X defined independently by Charalambous and Filippov, as well as, the relative inductive dimension defined by Chigogidze for a subspace X of a Tychonoff space Y may be considered as the base dimension I of X by normal bases Z ( X ) (all closed subsets of X), Z ( X ) (all functionally closed subsets of X), and Z ( X , Y ) = { X ∩ F : F ∈ Z ( Y ) } , respectively. In the present paper, we shall consider normal bases of spaces consisting of functionally closed subsets. In particular, we introduce new dimension invariant Ind 0 w : for a space X, Ind 0 w ( X ) is the minimal element α of the class O ∪ { − 1 , ∞ } , where O is the class of all ordinals, for which there exists a normal base C on X consisting of functionally closed subsets such that I ( X , C ) ⩽ α . We prove that in the class of all completely regular spaces X of weight less than or equal to a given infinite cardinal τ such that Ind 0 w ( X ) ⩽ n ∈ ω there exist universal spaces. However, the following questions are open. (1) Are there universal elements in the class of all normal (respectively, of all compact) spaces X of weight ⩽ τ with Ind 0 w ( X ) ⩽ n ∈ ω ? (2) Are there universal elements in the class of all Tychonoff (respectively, of all normal) spaces X of weight ⩽ τ with Ind 0 ( X ) ⩽ n ∈ ω ? (Note that Ind 0 w ( X ) = Ind 0 ( X ) for a compact space X.)