A construction of containing spaces

Topology and its Applications

2010

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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,In the present paper, we shall consider normal bases of spaces consisting of functionally closed subsets. In particular, we introduce new dimension invariant Ind w 0 : for a space X, Ind w 0 (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 w 0 (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 w 0 (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 w 0 (X) = Ind 0 (X) for a compact space X.)

Section: The Present Papermentioning

confidence: 99%

On the base dimension I and the property of universality

Topology and its Applications

2010

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“…In this paper we use all notions and notation introduced in [2]. In particular, if an indexed collection of spaces is denoted by the letter "S", a co-mark of S is denoted by the letter "M", and an M-admissible family of equivalence relations on S is denoted by the letter "R", then we always denote by "T" the containing space T(M, R) and by "B T " the standard base for T. If moreover M = {{U X δ : δ ∈ τ } : X ∈ S}, then the elements of B T are denoted by U T δ (H), δ ∈ τ and H ∈ C ♦ (R).…”

Section: Introductionmentioning

confidence: 99%

“…As in [2] we shall be concerned with classes, sets, collections, and families. A class is not necessarily a set.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Some properties of the containing spaces and saturated classes of spaces

2003

Appl. Gen. Topol.

Subjects of this paper are: (a) containing spaces constructed in [2] for an indexed collection S of subsets, (b) classes consisting of ordered pairs (Q,X), where Q is a subset of a space X, which are called classes of subsets, and (c) the notion of universality in such classes. We show that if T is a containing space constructed for an indexed collection S of spaces and for every X ϵ S, QX is a subset of X, then the corresponding containing space TIQ constructed for the indexed collection Q ={QX : X ϵ S} of spaces, under a simple condition, can be considered as a specific subset of T. We prove some “commutative” properties of these specific subsets. For classes of subsets we introduce the notion of a (properly) universal element and define the notion of a (complete) saturated class of subsets. Such a class is “saturated” by (properly) universal elements. We prove that the intersection of (complete) saturated classes of subsets is also a (complete) saturated class. We consider the following classes of subsets: (a) IP(Cl), (b) IP(Op), and (c) IP(n.dense) consisting of all pairs (Q;X) such that: (a) Q is a closed subset of X, (b) Q is an open subset of X, and (c) Q is a never dense subset of X, respectively. We prove that the classes IP(Cl) and IP(Op) are complete saturated and the class IP(n.dense) is saturated. Saturated classes of subsets are convenient to use for the construction of new saturated classes by the given ones.

“…In the present paper we shall use notions and notations of [7] (see also [5]) defined for different classes of T 0 -spaces of weight less than or equal to a given infinite cardinal τ . In particular, we shall use the notion of a Containing Space constructed for an arbitrary indexed collection of spaces.…”

mentioning

confidence: 99%

Universal Borel mappings and Borel actions of groups

Topology and its Applications

2008

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In this paper it is proved that for any two saturated (respectively, isometrically ω-saturated) classes (see [S.D. Iliadis, Universal Spaces and Mappings, North-Holland Mathematics Studies, vol. 198, Elsevier, 2005]) D and R of separable metrizable (respectively, separable metric) spaces and α ∈ ω + in the class of all Borel mappings of the class α whose domains belong to D and ranges to R there exist topologically (respectively, isometrically) universal elements. In particular, D and R can be independently one of the following saturated classes of separable metrizable (respectively, separable metric) spaces: (a) the class of all spaces, (b) the class of all countable-dimensional spaces, (c) the class of all strongly countable-dimensional spaces, (d) the class of all locally finite-dimensional spaces, (e) the class of all spaces of dimension less than or equal to a given non-negative integer, and (f) the class of all spaces of dimension ind less than or equal to a given non-finite countable ordinal. This result is not true if instead of the Borel mappings of the class α we shall consider the class of all Borel mappings.Using the construction of topologically (respectively, isometrically) universal mappings it is proved also that for an arbitrary considered separable metrizable group G and α ∈ ω + in the class of all G-spaces (X, F X ), where X belongs to a given saturated (respectively, isometrically ω-saturated) class P of spaces and the action F X of G on X is a Borel mapping of the class α, there exist topologically (respectively, isometrically) universal elements. In particular, P can be one of the above mentioned saturated (respectively, isometrically ω-saturated) classes of spaces. (About the notions of universality see below.) PreliminariesAgreement. All spaces are assumed to be separable metrizable. We shall consider also separable metric spaces. However, this fact will be explicitly verified.An ordinal is identified with the set of all smaller ordinals. A cardinal is identified with the first ordinal of the corresponding cardinality. By ω we denote the first infinite cardinal and by ω + the first cardinal larger than ω.The mappings are not necessary to be continuous. The domain of a mapping f is denoted by D f and the range by R f , that is if f is a mapping of a set X into a set Y , then D f = X and R f = Y .