The covering dimension invariants

In the present paper three types of covering dimension invariants of a space X are distinguished. Their sets of values are denoted by d - Sp U ( X ) , d - Sp W ( X ) and d - Sp β ( X ) . One of the exhibited relations between them shows that the minimal values of d - Sp U ( X ) , d - Sp W ( X ) and d - Sp β ( X ) coincide. This minimal value is equal to the dimension invariant mindim defined by Isbell. We show that if X is a locally compact space, then either d - Sp U ( X ) = [ mindim X , ∞ ] , or d - Sp U ( X ) = d - Sp β ( X ) = { dim X } . If X is not a pseudocompact space, then [ dim X , ∞ ] ⊂ d - Sp U ( X ) ; if X is a Lindelöff non-compact space, then d - Sp U ( X ) = [ dim X , ∞ ] ; if X is a separable metrizable non-compact space, then d - Sp W ( X ) = [ mindim X , ∞ ] . Among the properties of covering dimension invariants the generalization of the compactification theorem of Skljarenko is presented. The existence of compact universal spaces in the class of all spaces X with w ( X ) ⩽ τ and mindim X ⩽ n is proved.