Topological spaces connected to algebraic structures
by
A. V. Arhangelskii
Ohio University, Athens.

This talk is partially a survey and partially a set of new results and problems. Topological algebra and its objects may be of special interest to general topologists for several reasons. We mention below just two of them. First, the presence of an algebraic structure nicely related to a topology changes dramatically relationship between topological invariants. Important classical results in thus direction are well known; enough to mention Birkhgof-Kakutani's theorem that first countability is equivalent to metrizability in topological groups, Pontryagin's theorem that every topological group is a Tychonoff space, and Bourbaki's theorem that every locally compact topological group is paracompact. We present below some recent results of this kind and formulate a few challenging open problems.

Recall that a paratopological group is a group with a Hausdorff topology such that the multiplication is jointly continuous. First-countable paratopological groups, unlike topological groups, needn't be metrizable. However, it turns out that the existence of G_d -points in paratopological groups is intimately related to submetrizability. Some partial results related to the following major open problem are given below.

Problem 1. Is every regular first-countable paratopological group submetrizable?

Pontryagin's theorem also does not generalize to paratopological groups. However, the next question is open.

Problem 2. Is every regular paratopological group Tychonoff?

The second reason for a general topologist to be interested in topological algebra lies in the following. The structures of topological algebra naturally produce some standard constructions unavailable in pure general topology. These constructions lead to algebraico-topological objects which turn out to be highly non-trivial when treated as topological spaces. An important construction of this kind is that of the free topological group F(X) over a Tychonoff space X.

We may apply classical constructions of general topology (like that of the Stone-Cech compactification) to various concrete objects of topological algebra, which further expands the range of topological spaces within our reach.

In particular, we propose to systematically consider the remainders of topological groups in Hausdorff compactifications. Of course, these spaces, "generated" by topological groups, do not inherit from them any algebraic structure.

Example 1. Let Seq be the simplest infinite compact space, that is, the Alexandroff one-point compactification of the discrete space of natural numbers. Denote by F(Seq) the free topological group of the space Seq, and let Z be the Stone-Cech remainder of the space F(Seq). Then Z has the following curious collection of properties:

1) Z is nowhere locally compact.

2) Z is Cech-complete, and the Souslin number of Z is countable. Moreover, w₁ is a precaliber of Z.

3) Z is not paracompact.

4) The closure in Z of every s-compact subset of Z is compact; Hence, Z is not separable. However, Z contains a dense Lindelöf Cech-complete subspace.

This example shows that the Stone-Cech remainder of a quite nice, countable and sequential, topological group may be a space of a very different kind, with an unusual combination of properties.

In general, the properties of remainders of topological groups are not yet well understood. However, we should expect much stronger connections between the properties of the remainders and the properties of the groups than in the general case of arbitrary Tychonoff spaces. In this direction we mention the following results:

Theorem 1. [1] Suppose that G is a topological group and that M is a remainder of G (in a Hausdorff compactification of G). Then the following conditions are equivalent to each other:

a) M is submetrizable;

b) M is metrizable;

c) M has a point-countable base.

Besides, if G is not locally compact, then each of these conditions implies that G is separable and metrizable.

Here is a partial strengthening of the above theorem:

Theorem 2. If a non-locally compact topological group G has a remainder which is the union of a finite collection of metrizable subspaces, then G is metrizable.

In the last theorem it is enough to assume that the remainder is the union of a finite collection of subspaces with a point-countable base. In fact, we can just assume that the remainder is the union of finitely many of first-countable hereditarily D-spaces.

With the help of Theorem 1 the next result can be established (see also [1]).

Theorem 3. (CH) Suppose that X is a compact Hausdorff space such that |X| ≤ 2^w . Suppose also that X=Y∪Z, where Y and Z are homeomorphic to (possibly, different) topological groups. Then Y and Z are metrizable, and if none of them is locally compact, then both Y and Z are separable.

Observe that we do not suppose that Y and Z are disjoint.

Example 2. The double arrow space Dar of Alexandroff and Urysohn is the union of two disjoint copies of Sorgenfrey line S. Notice, that S is a paratopological group, and that Dar is a compact Hausdorff space of the cardinality 2^w . Thus, the last theorem does not generalize to the unions of paratopological groups.

Problem 3. When a topological group G has a paracompact remainder? Must G itself be paracompact in this case? Must G be Dieudonne complete?

(Give necessary conditions, sufficient conditions, and non-trivial examples).

Problem 4. When a topological group G has a remainder with the Baire property?

In connection with the last question, we mention the next easy to establish fact:

Proposition. If a topological group G is either pseudocompact or s-compact, then every remainder of G in a Hausdorff compactification has the Baire property.

References

[1] Arhangel'skii, A.V., More on remainders close to metrizable spaces, Topology and Appl. 154 (2007), 1084-1088.

Date received: July 13, 2007

Copyright © 2007 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 # cavh-18.