Some properties of the containing spaces and saturated classes of spaces

Stavros Iliadis

Abstract

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.


Keywords

Containing space; Universal space; Saturated class of spaces; Saturated class of subsets

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References

R. Engelking, W. Holsztynski and R. Sikorski, Some examples of Borel sets, Colloq. Math. 15 (1966), 271-274. https://doi.org/10.4064/cm-15-2-271-274

S. D. Iliadis, A construction of containing spaces, Topology Appl. 107 (2000), 97-116. https://doi.org/10.1016/S0166-8641(00)90095-6

R. Sikorski, Some examples of Borel sets, Colloq. Math. 5 (1958), 170-171. https://doi.org/10.4064/cm-5-2-170-171

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1. On the base dimension I and the property of universality
Stavros Iliadis
Topology and its Applications  vol: 157  issue: 4  first page: 752  year: 2010  
doi: 10.1016/j.topol.2009.08.027



Esta revista se publica bajo una licencia de Creative Commons Reconocimiento-NoComercial-SinObraDerivada 4.0 Internacional.

Universitat Politècnica de València

e-ISSN: 1989-4147   https://doi.org/10.4995/agt