Countable Topological Group Research Articles

Discrete Subsets in Topological Groups and Countable Extremally Disconnected Groups

It is proved that any countable topological group in which the filter of neighborhoods of the identity element is not rapid contains a discrete set with precisely one nonisolated point. This gives a negative answer to Protasov's question on the existence in ZFC of a countable nondiscrete group in which all discrete subsets are closed. It is also proved that the existence of a countable nondiscrete extremally disconnected group implies the existence of a rapid ultrafilter and, hence, a countable nondiscrete extremally disconnected group cannot be constructed in ZFC.

Generating countable groups by discrete subsets

Every countable topological group G has a closed discrete subset A such that G=AA−1.

Selective covering properties of product spaces, II: $\gamma $ spaces

We study productive properties of gamma spaces, and their relation to other, classic and modern, selective covering properties. Among other things, we prove the following results: 1. Solving a problem of F. Jordan, we show that for every unbounded tower set of reals X of cardinality aleph_1, the space Cp(X) is productively FU. In particular, the set X is productively gamma. 2. Solving problems of Scheepers and Weiss, and proving a conjecture of Babinkostova-Scheepers, we prove that, assuming CH, there are gamma spaces whose product is not even Menger. 3. Solving a problem of Scheepers-Tall, we show that the properties gamma and Gerlits--Nagy (*) are preserved by Cohen forcing. Moreover, every Hurewicz space that Remains Hurewicz in a Cohen extension must be Rothberger (and thus (*)). We apply our results to solve a large number of additional problems, and use Arhangel'skii duality to obtain results concerning local properties of function spaces and countable topological groups.

On topological groups with a first-countable remainder, II

We establish estimates on cardinal invariants of an arbitrary non-locally compact topological group G with a first-countable remainder Y. We show that the weight of G and the cardinality of Y do not exceed 2ω. Moreover, the cardinality of G does not exceed 2ω1. These bounds are best possible as witnessed by a single topological group G. We also prove that every precompact topological group with a first-countable remainder is separable and metrizable. It is known that under Martin's Axiom and the negation of the Continuum Hypothesis, every σ-compact topological group with a first-countable remainder is metrizable. We show that under the Continuum Hypothesis, there is an example of a countable topological group which is not metrizable and has a first-countable remainder. Hence for countable groups, the question of whether the existence of a first-countable remainder is equivalent to being metrizable, is undecidable.

Equivariant principal bundles and their classifying spaces

We consider \Gamma-equivariant principal G-bundles over proper \Gamma-CW-complexes with prescribed family of local representations. We construct and analyze their classifying spaces for locally compact, second countable topological groups with finite covering dimension \Gamma and G such that G is almost connected.

What is typical?

Let ξ be a random measure on a locally compact second countable topological group, and let X be a random element in a measurable space on which the group acts. In the compact case we give a natural definition of the concept that the origin is a typical location for X in the mass of ξ, and prove that when this holds, the same is true on sets placed uniformly at random around the origin. This new result motivates an extension of the concept of typicality to the locally compact case where it coincides with the concept of mass-stationarity. We describe recent developments in Palm theory where these ideas play a central role.

What is typical?

Let ξ be a random measure on a locally compact second countable topological group, and letXbe a random element in a measurable space on which the group acts. In the compact case we give a natural definition of the concept that the origin is a typical location forXin the mass of ξ, and prove that when this holds, the same is true on sets placed uniformly at random around the origin. This new result motivates an extension of the concept of typicality to the locally compact case where it coincides with the concept of mass-stationarity. We describe recent developments in Palm theory where these ideas play a central role.

Generically there is but one self homeomorphism of the Cantor set

We describe a self homeomorphism R of the Cantor set X and then show that its conjugacy class in the Polish group H(X) of all homeomorphisms of X forms a dense G δ subset of H(X). We also provide an example of a locally compact, second countable topological group which has a dense conjugacy class.

Local Homomorphisms of Topological Groups

AbstractA mapping f : G → s from a left topological group G into a semigroup S is a local homomorphism if for every x є G \ {e}, there is a neighborhood Ux of e such that f (xy) = f (x)f (y) for all y є Ux \ {e}. A local homomorphism f : G → S is onto if for every neighborhood U of e, f(U \ {e}) = S. We show that(1) every countable regular left topological group containing a discrete subset with exactly one accumulation point admits a local homomorphism onto N,(2) it is consistent that every countable topological group containing a discrete subset with exactly one accumulation point admits a local homomorphism onto any countable semigroup,(3) it is consistent that every countable nondiscrete maximally almost periodic topological group admits a local homomorphism onto the countably infinite right zero semigroup.

On partitions of groups into dense subsets

It is proved that every countable topological group not containing an open Boolean subgroup can be partitioned into countably many dense subsets. It is also proved that every countable group with finitely many elements of order 2, that can be embedded in a compact topological group, can be partitioned into countably many subsets dense in any nondiscrete group topology.

Homogeneous spaces of curvature bounded below

We prove that every locally connected quotient G/H of a locally compact, connected, first countable topological group G by a compact subgroup H admits a G-invariant inner metric with curvature bounded below. Every locally compact homogeneous space of curvature bounded below is isometric to such a space. These metric spaces generalize the notion of Riemannian homogeneous space to infinite dimensional groups and quotients which are never (even infinite dimensional) manifolds. We study the geometry of these spaces, in particular of non-negatively curved homogeneous spaces.

Joint continuity of separately continuous mappings on topological groups

The main theorem of this paper is somewhat stronger than the following statement: Let G G be a Baire semitopological group, let H H be a first countable one and let N N be a first countable topological group; then each separately continuous bi-homomorphism from G × H G\times H into N N is jointly continuous. This theorem has some consequences on joint continuity of separately continuous multiplication of rings and scalar multiplication of modules.

Some nonmetric, first countable, cancellative topological semigroups that are generalized metric spaces

A first countable topological group is metrizable. Certain nonfirst countable topological groups are “generalized metric spaces”. To elucidate the latter, we briefly list some theorems and examples. To indicate the difficulty of generalizing those theorems (especially the first countable case) even to “nice” topological semigroups we exhibit some nonmetrizable first countable cancellative topological semigroups that are, respectively: a quasi-metrizable, locally compact, Moore space; a nonquasi-metrizable Moore space; a Nagata space having no semimetric that is separately continuous in one variable; and a paracompact, quasi-metrizable, quasi-developable space that is neither developable nor stratifiable.

A Borel Parametrlzatlon of Polish Groups

This paper constructs a standard space, PG, and a map p^PG—>G(p} such that each G(p) is a Polish group, and such that every Polish group is isomorphic to at least one of the groups G(p) ; PG thus serves as a parameter space for all Polish groups. We formulate the notion of a map from a standard B Space to Polish groups, and that of a from a standard groupoid to Polish groups; both are defined in terms of the existence of factorizations through PG. We apply these ideas to establish a general Cohomology Lemma, asserting that cocycles, with values in family of Polish groups, may be cobounded into a given family of dense, normal, subgroups, whenever the underlying groupoid is a hyperfinite equivalence relation. The purpose of this paper is to provide a parametrizatio n, by a standard space PG, for the space of Polish topological groups, i. e. those second countable topological groups whose underlying topology may be defined by a complete metric, and to present applications of this to the notion of Borel functor from a standard groupoid to Polish groups. The need for such concepts became apparent during the course of joint work with M. Takesaki on the classification of the possible actions (up to cocycle conjugacy) of a discrete amenable group on a hyperfinite, semifinite injective von Neumann algebra [12], and the paper can be viewed as preparatory to this work. However, the point of view adopted also reveals a definition of A. Connes, [4], of the notion of from a standard groupoid to standard measure spaces, as being very natural. A common situation in which the problems considered here arise is the following: if G is a Polish group and X a Polish G-space under

Metrizability and the Fréchet-Urysohn property in topological groups

A question of Arhangel’skii, whether weakly first countable topological groups are metrizable, is answered in two ways: if the Hausdorff axiom is assumed, the answer is yes, but in general a weakly first countable topological group need not be pseudometrizable. The former result is obtained as a corollary of a more general sufficient condition for a sequential group to be Fréchet-Urysohn. A general necessary and sufficient condition for a sequential group to be Fréchet-Urysohn is given, and a number of questions are raised. Examples are given to show in what respect the theorems of the paper are the "best possible".

On the Measurability of Orbits in Borel Actions

We replace measure with category in an argument of G. W. Mackey to characterize closed subgroups H of a totally nonmeager, 2nd countable topological group G in terms of the quotient Borel structure G/H. As a corollary, we obtain an improved version of a theorem of C. Ryll-Nardzewski on the Borel measurability of orbits in continuous actions by Polish groups.

On the measurability of orbits in Borel actions

We replace measure with category in an argument of G. W. Mackey to characterize closed subgroups H of a totally nonmeager, 2nd countable topological group G in terms of the quotient Borel structure G/H. As a corollary, we obtain an improved version of a theorem of C. Ryll-Nardzewski on the Borel measurability of orbits in continuous actions by Polish groups.

The Baire set fixing property in topological groups

A now standard technique for treating a class of Baire set problems, originating with Halmos [2; 51.D], is to construct a metric space that, in a way, "fixes" a given Baire set. Formally this means, given a Baire set B in a space X, find a metric space M and a map g of X onto M with the property that B = g l gB . In phrasing the problem for topological groups it is natural to require that M be a group and g be a homomorphism. Specifically, what is desired is a subgroup H that "fixes" B, i.e., such that B = BH. This problem is solved in Halmos for locally compact a-compact groups in a theorem [2; 64.G] that has led to interesting consequences elsewhere [4, 1]. As will be seen, Halmos' theorem is also true for totally bounded groups. An example of Sirota's of a countable topological group with no non-trivial metrizable homomorphic images indicates that, in this direction at least, the barrier of local compactness cannot be lifted without giving up metrizability in M. If this is done, however, replacing the requirement that M be metrizable, i.e., that H have a countable base in G, with the less stringent requirement that H be a G~-set in G, an analogue of Halmos' theorem is true for Lindelff groups. In view of these remarks it is convenient to distinguish between two distinct Baire set fixing problems. A closed invariant subgroup H of a group G is said to be a fixiru3 subgroup (M-fixing subgroup) provided H is a G~-set (H has a countable base of neighborhoods in G). G satisfies the Baire set fixing property (M-Baire set fixing property) provided, for each Baire set in G, there is a fixing subgroup (M-fixing subgroup) such that B = BH. The Baire sets are taken to be the elements of the smallest a-algebra containing the zero sets of continuous real-valued functions on G. For technical convenience in describing proofs, a G~-set containing the identity is called an admissible set.