Metrizability of CHART groups

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Evgenii Reznichenko

Department of General Topology and Geometry, Mechanics and Mathematics Faculty,

M. V. Lomonosov Moscow State University, Leninskie Gory 1, Moscow, 199991 Russia

Abstract

For compact Hausdorﬀ admissible right topological (CHART) group 𝐺, we prove

𝑤(𝐺) = 𝜋𝜒(𝐺). This equality is well known for compact topological groups.

This implies the criteria for the metrizability of CHART groups: if 𝐺is ﬁrst-

countable (2013, Moors, Namioka) or 𝐺is Fr´echet (2013, Glasner, Megrelishvili),

or 𝐺has countable 𝜋-character (2022, Reznichenko) then 𝐺is metrizable. Under

the continuum hypothesis (CH) assumption, a sequentially compact CHART

group is metrizable. Namioka’s theorem that metrizable CHART groups are

topological groups extends to CHART groups with small weight.

Keywords: compact right topological groups, admissible groups, CHART

groups, metrizable spaces, 𝜋-character

1. Introduction

A group 𝐺with a topology is called right topological if all right shifts 𝜌ℎ:

𝐺→𝐺, 𝑔 ↦→ 𝑔ℎ are continuous. The set of 𝑔∈𝐺for which the left shift

𝜆𝑔:𝐺→𝐺, ℎ ↦→ 𝑔ℎ is continuous is called the topological center and is denoted

as Λ(𝐺). A group 𝐺with topology is called semitopological if all right and

left shifts are continuous, that is, if 𝐺is right topological and Λ(𝐺) = 𝐺. A

right topological group 𝐺is called admissible if Λ(𝐺)is a dense subset of 𝐺.

We write “CHART” for “compact Hausdorﬀ admissible right topological”. A

paratopological group 𝐺is a group 𝐺with a topology such that the product

map of 𝐺×𝐺into 𝐺is jointly continuous.

The study of groups with topology in which not all operations are continuous

and conditions implying the continuity of operations began with the 1936 paper

[1] of Montgomery, who, among other things, proved that a Polish (i.e., sepa-

rable metrizable by a complete metric) semitopological group is a topological

group. Interest in such groups was renewed in relation to topological dynam-

ics. The autohomemorphism group of a locally compact space in the compact

open topology is a paratopological group [2]. In the same paper Arens obtained

conditions under which an autohomemorphism group is a topological group. In

Email address: erezn@inbox.ru (Evgenii Reznichenko)

Preprint submitted to Elsevier January 16, 2023

arXiv:2210.01631v3 [math.GR] 13 Jan 2023

1957 Ellis proved that a locally compact paratopological group is a topological

group [3]. Shortly afterwards, he strengthened this theorem to semitopologi-

cal locally compact groups and proved the celebrated Ellis theorem [4]: any

semitopological locally compact group is a topological group. The enveloping

semigroup of a dynamical system was introduced by Ellis in 1960 [5]. It has

become a fundamental tool in the abstract theory of topological dynamical sys-

tems. The study of dynamical systems for which the enveloping semigroup is a

CHART group plays a large role in the abstract theory of topological dynamical

systems.

Let 𝐺be a CHART group. Any of the following conditions implies that 𝐺

is a topological group.

(𝐶1)𝐺is metrizable (Theorem 2.1 [6]).

(𝐶2)𝐺is ﬁrst-countable (Remark after Proposition 2.7 [7]).

(𝐶3)𝐺is Fr´echet (Corollary 8.8 [8]).

(𝐺4)𝐺has countable 𝜋-character (for example, 𝐺is a compact space with

countable tightness) (Corollary 2 (3) [9]).

A CHART group 𝐺is tame if for every 𝑔∈𝐺, the mapping 𝑥↦→ 𝑔·𝑥is

fragmented. A compact semitopological group is tame.

Note even more conditions that imply that 𝐺is a topological group: (𝐺5)the

multiplication of 𝐺is separately continuous (Ellis theorem [4]); (𝐺6)𝐺is tame

[8, Theorem 21]; (𝐺7)the multiplication of 𝐺is continuous at (𝑒, 𝑒); (follows

from [10] and [11, Theorem 5]); (𝐺8)the multiplication of 𝐺is feebly continuous

[12, Proposition 3.2] (see also [9, Corollary 2 (2)]); (𝐺9)the inversion 𝑔↦→ 𝑔−1

is continuous at 𝑒[11, Theorem 5]; (𝐺10) Λ(𝐺)is a topological group (or merely

contains a dense topological group) [11, Theorem 5]; the right translations 𝑔↦→

𝑔ℎ form an equicontinuous family of maps from 𝐺onto 𝐺[11, Theorem 5].

Note that the group 𝐺in (𝐶1)–(𝐶4)is metrizable because compact ﬁrst-

countable, Fr´echet topological groups and groups of countable 𝜋-character are

metrizable (Corollary 4.2 .2 and Corollary 5.7.26 of [13]). Recall that for com-

pact Hausdorﬀ spaces [14]:

metrizable ⇒ﬁrst-countable ⇒Fr´echet ⇒

countable tightness ⇒countable 𝜋-character.

In this note, we prove that 𝑤(𝐺) = 𝜋𝜒(𝐺)for CHART group 𝐺(Theorem

2). Whence it follows that 𝐺is metrizable if 𝐺has countable 𝜋-character. Note

that this fact reduces (𝐶2)–(𝐶4)to (𝐶1).

Under the continuum hypothesis (CH) assumption, a sequentially compact

CHART group is metrizable (Corollary 2). Namioka’s theorem that metrizable

CHART groups are topological groups extends to CHART groups with small

weight (Theorem 5).

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摘要：

MetrizabilityofCHARTgroupsEvgeniiReznichenkoDepartmentofGeneralTopologyandGeometry,MechanicsandMathematicsFaculty,M.V.LomonosovMoscowStateUniversity,LeninskieGory1,Moscow,199991RussiaAbstractForcompactHausdorffadmissiblerighttopological(CHART)group�,weprove�(�)=&#...

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Metrizability of CHART groups

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