Moscow spaces and their applications
by
Alexander Arhangel'skii
Ohio University

Moscow spaces were introduced in [1]. Their definition is very elementary. A space X is called Moscow if, for each open subset U of X, the closure of U is the union of a family of G_\delta -subsets of X. Recently Moscow spaces found interesting and far reaching applications in topological algebra, mostly to the theory of topological groups. These applications are based on certain curious purely topological properties of Moscow spaces themselves, and on the fact that topological spaces with a mild algebraic structure turn out to be Moscow amazingly often.

Our goal in the talk is threefold:

1. To present some applications of Moscow spaces to the following questions:

   1. When the equality \nuX×\nuY=\nu(X×Y) holds, where \nuX is the Hewitt-Nachbin completion (that is, realcompactification) of X?
   2. When a space X is power homogeneous, that is, there exists a cardinal number \tau such that X^\tau is homogeneous?
   3. When, for a topological group G, the operations can be continuously extended to the Hewitt-Nachbin completion \nuG of the space G?
2. To show how large is the class of Moscow spaces, and to establish that topological groups and semitopological groups are even more often Moscow than topological spaces in general (see [2]). Note, that the class of Moscow spaces contains, on one hand, the class of all spaces of countable pseudocharacter and, on the other hand, the class of all extremally disconnected spaces.
3. To discuss some basic properties of Moscow spaces, in particular, those which are especially important for applications.

Many new open questions are formulated; some of them are closely related to certain difficult old problems in general topology and topological algebra.

References

[1] Arhangel'skii A.V., Functional tightness, Q-spaces, and \tau-embeddings, Comment. Math. Univ. Carolinae 24:1 (1983), 105-120.

[2] Arhangel'skii A.V., On a Theorem of W.W. Comfort and K.A. Ross, Comment. Math. Univ. Carolinae 40:1 (1999), 133-151.

Date received: June 28, 2000

Copyright © 2000 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 Conferences Inc. Document # caeh-48.