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Extensions of Topological and Semitopological Groups and the Product Operation Commentationes Mathematicae Universitatis Carolinae Aleksander V. Arhangel'skii; Miroslav Hušek Extensions of topological and semitopological groups and the product operation Commentationes Mathematicae Universitatis Carolinae, Vol. 42 (2001), No. 1, 173--186 Persistent URL: http://dml.cz/dmlcz/119232 Terms of use: © Charles University in Prague, Faculty of Mathematics and Physics, 2001 Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz Comment.Math.Univ.Carolin. 42,1 (2001)173–186 173 Extensions of topological and semitopological groups and the product operation A.V. Arhangel’skii, M. Huˇsek Abstract. The main results concern commutativity of Hewitt-Nachbin realcompactifica- tion or Dieudonn´ecompletion with products of topological groups. It is shown that for every topological group G that is not Dieudonn´ecomplete one can find a Dieudonn´e complete group H such that the Dieudonn´ecompletion of G × H is not a topological group containing G×H as a subgroup. Using Korovin’s construction of Gδ-dense orbits, we present some examples showing that some results on topological groups are not valid for semitopological groups. Keywords: topological group, Dieudonn´ecompletion, PT-group, realcompactness, Moscow space, C-embedding, product Classification: 22A05, 54H11, 54D35, 54D60 §0. Introduction Although many of our general results are valid for more general spaces, we shall assume that all the spaces under consideration are Tychonoff. The following notion was introduced in [Ar1]. A space X is called Moscow, if, for each open subset U of X, the closure of U in X is the union of a family of Gδ-subsets of X, that is, for each x ∈ U¯ there exists a Gδ-subset P of X such that x ∈ P ⊂ U¯. A topological group is called Moscow if it is a Moscow space. The techniques based on the notion of Moscow space played a vital role in the recent solution of the next problem posed by V.G. Pestov and M.G. Tkaˇcenko in [PT]. They asked the following question, which we call below the PT-problem. Let G be a topological group, and µG the Dieudonn´ecompletion of the space G. Can the operations on G be extended to µG in such a way that µG becomes a topological group, containing G as a topological subgroup? That means, is µG a topological group such that the reflection map G → µG is a homomorphism? Recall that the Dieudonn´ecompletion µG of G is the completion of G with respect to the maximal uniformity on G compatible with the topology of G. In other words, it is a reflection of G in the smallest productive and closed heredi- tary class containing all metrizable spaces. It is well known that the Dieudonn´e completion of a topological space X is always contained in the Hewitt-Nachbin The second author acknowledges support of the grants GACRˇ 201/00/1466 and MSM 113200007. 174 A.V. Arhangel’skii, M. Huˇsek completion υX of X (the smallest productive and closed hereditary class contain- ing the space of reals). Clearly, µX is the smallest Dieudonn´ecomplete subspace of υX containing X. Moreover, if there are no Ulam-measurable cardinals, then υX and µX coincide by Shirota theorem ([Sh]). Therefore, the next question, also belonging to Pestov and Tkaˇcenko (see [Tk]), is almost equivalent to the above question. Let G be a topological group, and let υG be the Hewitt-Nachbin completion of the space G. Can the operations on G be extended to υG in such a way that υG becomes a topological group, containing G as a topological subgroup? A counterexample to the PT-problem was described in [Ar2]. We call a topolog- ical group G a PT-group if µG is a topological group, containing G as a topological subgroup. It was shown in [Ar2] that every Moscow group is a PT-group. One of the main questions we consider in this note is as follows: when the product of PT-groups is a PT-group? We also consider some open questions concerning various natural extensions of topological groups G. In what follows ρG is the Rajkov completion of G and ρωG is the Gδ-closure of G in ρG (see [RD]), that is, the set of all points in ρG which can not be separated from G by a Gδ subset of ρG. Clearly, ρωG is also a topological subgroup of ρG. The problems we deal with deeply involve the notion of C-embedding and are all closely related to the PT-problem. Some results we formulate for more general situations. Recall that a group endowed with a topology is called a right topological group if the binary group operation is continuous in the left variable, when the right one is fixed; a semitopological group if the binary group operation is separately continuous; a paratopological group if the binary group operation is jointly continuous; a quasitopological group if the binary group operation is separately continuous and the inversion is continuous; a topological group if the binary group operation is jointly continuous and the inversion is continuous. The first two concepts and terms are used also for the case when G is a semi- group. As for the third concept, a semigroup having its binary operation contin- uous is called a topological semigroup. Although it is not always needed, we shall assume that semigroups have units and homomorphisms preserve units. Note that, unlike in right topological semigroups, the translations in right topological groups are homeomorphisms (and the spaces are homogeneous). Nature of some of the next results for µX is categorical and we shall formulate them for a general reflection cX of a topological space X (in fact, c need not be defined on all the Tychonoff spaces, it suffices it is defined on the spaces under consideration). In most situations, X is densely embedded into cX (like βX or υX or µX for Tychonoff spaces X). There are also some natural situations where Extensions of topological and semitopological groups and the product operation 175 the reflection map sends X onto a dense subspace of cX but the map is not one- to-one (like zerodimensional compactifications). Other situations do not happen too often (like Herrlich’s example of reflections into powers of a strongly rigid space). Take a family {Xi}i∈I of nonvoid topological spaces. Denote by p the canonical continuous map c(ΠXi) → ΠcXi that is a reflective extension of the product of reflection maps ΠXi → ΠcXi. It is also generated (as a mapping into a product) by the extensions c(prj) : c(ΠXi) → cXj of prj : ΠXi → Xj for j ∈ I. Since Xj is a retract of ΠXi, the reflection cXj is a retract of c(ΠXi), where the retraction is the composition of p and of the projection onto cXj. In case when spaces under consideration are densely embedded in their c-reflections, the restriction of p to the closure in c(ΠXi) of Xj ×{a} (where a ∈ Πi∈I\{j}Xi) is a homeomorphism onto cXj ×{a}. §1. Products of extensions If we say for a group (or semigroup) X endowed with a topology that cX is a group (or semigroup, resp.), we also assume that the corresponding reflection map X → cX is a homomorphism. Clearly, if c(ΠXi) is a semitopological group (or semigroup), then cXi are groups (or semigroups, resp.), too (using extensions of the embeddings Xj → Xj × {e} ⊂ ΠXi and then the above retractions). Moreover, then the map p is a homomorphism; we can show it for dense embedding reflections (the general case has the same idea but is formally more complicated): the continuous maps p(x · y) and p(x) · p(y) for a fixed x ∈ ΠXi coincide on the dense subset ΠXi, so they are equal, and now repeat the same procedure for the same maps for fixed y ∈ c(ΠXi). We conjecture that the situation above does not hold for right topological groups although the next result would suggest its more general application than for semitopological groups. Lemma 1.1. Let G1 and G2 be right topological groups, and let H be a dense subspace of G1. Suppose further that f : G1 → G2 is a continuous homomor- phism of G1 into G2 such that the restriction of f to H is a topological embedding. Then f is a topological embedding of G1 into G2. Proof: Assume the contrary. Then there exist x ∈ G1 and A ⊂ G1 such that x is not in the closure of A, while y = f(x) is in the closure of B = f(A). Since the natural translations in G1 and G2 commute with the homomorphism f, we may assume that x belongs to H. Since the space G1 is regular, we may also assume that A ⊂ H. However, now the contradiction is obvious: since f|H is a homeomorphism of H onto f(H), and x ∈ H, A ⊂ H, it follows that f(x) is not in the closure of f(A). As a direct consequence of the above lemma, we get the following easy but important result. Since we do not know whether p is a homomorphism for right 176 A.V. Arhangel’skii, M. Huˇsek topological groups, we have to formulate it for semitopological groups. See The- orem 2.8, where also the converse statement is proved for c = µ.
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