Dimension-like functions and universality

摘要

In the paper [A.K. O'Connor, A new approach to dimension, Acta Math. Hungar. 55 (1-2) (1990) 83-95] two dimensions, denoted by dm and Dm, are defined in the class of all Hausdorff spaces. The dimension Dm does not have the universality property in the class of separable metrizable spaces because the family of all such spaces X with Dm(X) ≤ 0 coincides with the family of all totally disconnected spaces in which there are no universal elements (see [R. Pol, There is no universal totally disconnected space, Fund. Math. 79 (1973) 265-267]). In the present paper modifications of dm and Dm are given in order to obtain new dimension-like functions (briefly, dimensions) having the universality property. These new dimensions are defined in the class of T_0-spaces and denoted by dm_E~(K,B) and _E~(K,B), where E is a class of spaces, K is a class of subsets, and B is a class of bases. We prove that if K, B, and E are saturated (see [S.D. Iliadis, Universal Spaces and Mappings, North-Holland Mathematics Studies, vol. 198, Elsevier Science B.V., Amsterdam, 2005, xvi+559 pp.]), then for a given saturated class P of spaces and non-negative integer k in the family of all spaces X of P such that dm_E~(K,B) (X) ≤ κ (respectively, Dm_E~(K,B) (X) ≤ κ) there exist universal elements. We recall (see [S.D. Iliadis, Universal Spaces and Mappings, North-Holland Mathematics Studies, vol. 198, Elsevier Science B.V., Amsterdam, 2005, xvi+559 pp.]) that for a fixed infinite cardinal τ the classes P of (a) T_0-spaces of weight ≤ τ, (b) (completely) regular spaces of weight ≤ τ, (c) (completely) regular countable-dimensional spaces of weight ≤ τ, (d) (completely) regular strongly countable-dimensional spaces of weight ≤ τ, (e) (completely) regular locally finite-dimensional spaces of weight ≤ τ, and of (f) (completely) regular spaces X of weight ≤ τ with ind(X) ≤ α ∈ τ~+ are saturated.