Some properties of the containing spaces and saturated classes of spaces

Authors

  • Stavros Iliadis University of Patras

DOI:

https://doi.org/10.4995/agt.2003.2047

Keywords:

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

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.

Downloads

Download data is not yet available.

Author Biography

Stavros Iliadis, University of Patras

Department of Mathematics

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

Downloads

Published

2003-10-01

How to Cite

[1]
S. Iliadis, “Some properties of the containing spaces and saturated classes of spaces”, Appl. Gen. Topol., vol. 4, no. 2, pp. 487–507, Oct. 2003.

Issue

Section

Articles