Sequential Envelope Revisited
by
Roman Fric
Slovak Academy of Sciences
Coauthors: Nobuyuki Kemoto (Oita University)

Traditionally, the extension of (bounded) continuous real-valued functions in topology is associated with the Cech-Stone compactification and the Hewitt realcompactification. It has been generalized to E-compactifications by S. Mrówka and all such constructions are particular cases of epireflections in categorical topology.

Our talk is devoted to sequential envelopes - the sequential counterpart to the topological extensions. We concetrate on two topics. 1. A modification of the sequential envelope applicable to the foundation of Probability (Section 2). 2. Sequential completeness - the property of spaces with empty sequential growth (Section 3).

The idea that the extension of probability measures from a field of events to the generated \sigma-field is of a topological nature was presented by J. Novák in 1954 at a meeting on Probability and Statistics in Berlin and in 1950 at the III. Mathematical Congress in Moscow (see NOVÁK [1958]). The outcome is the theory of sequential envelopes outlined at the First Prague Topological Symposium in 1961 (see NOVÁK [1962]) and presented in NOVÁK [1965] and [1968]. It was M. Husek who in 1971 (an unpublished manuscript) pointed out the categorical background of the sequential envelope: it is an epireflection of a sequential convergence space belonging to the category simply generated by the closed interval [0, 1] or the real line R to the subcategory of absolutely sequentially closed spaces. Even though the categorical description of the sequential envelope is that simple, the story was not quite straightforward and some results were certainly surprising for the founder of the theory of sequential envelopes.

The aim of the present talk is to survey briefly some published results and examples related to sequential envelopes and also to present some new results. We believe that there are still some surprises and tough problems concerning sequential envelopes ahead ...

In Section 1 we analyse the notion of sequential envelope \sigma from the point of view of its application to the foundation to Probability. In Section 2 we describe its modification called H-completion; for H = Z(2) we get exactly the generated \sigma-field of events as the Z(2)-completion of a field of events. Seciton 3 is devoted to sequentially complete spaces, i.e. spaces X for which ``X = \sigmaX''.

We have tried to suppress technical details about \Cal L-spaces and closure spaces (as used in NOVÁK [1962], [1965], [1968] and to use as much topological language as possible. To make our presentation more self-contained, we have added a short Appendix containing a description of some basic sequential convergence notions.

Written during the study stay in Japan supported by the Japan Society for the Promotion of Science; partially supported by Grant GA SAV 1230/96

Date received: June 24, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts. Document # caaj-28.