Countably Compact Group Research Articles

Real-valued measurable cardinals and sequentially continuous homomorphisms

A. V. Arkhangel'skiĭ asked in 1981 if the variety V of topological groups generated by free topological groups on metrizable spaces coincides with the class of all topological groups. We show that if there exists a real-valued measurable cardinal then the variety V is a proper subclass of the class of all topological groups. A topological group G is called g-sequential if for any topological group H any sequentially continuous homomorphism G→H is continuous. We introduce the concept of a g-sequential cardinal and prove that a locally compact group is g-sequential if and only if its local weight is not a g-sequential cardinal. The product of a family of non-trivial g-sequential topological groups is g-sequential if and only if the cardinal of this family is not g-sequential. Suppose G is either the unitary group of a Hilbert space or the group of all self-homeomorphisms of a Tikhonov cube. Then G is g-sequential if and only if its weight is not a g-sequential cardinal. Every compact group of Ulam-measurable cardinality admits a strictly finer countably compact group topology.

Some generalized countably compact properties in topological groups

In this paper, firstly, we study strict q-spaces, strong q-spaces and point pseudocompact spaces in topological groups in terms of preimages of metrizable spaces. A continuous mapping f:X→Y is (strongly) sequential-perfect if f is closed and (f−1(F)) f−1(y) is a sequentially compact set for each (sequentially compact set F⊆Y) y∈Y. We give some characterizations as follows:(1)A topological group G is a strict q-space if and only if it is a sequential-perfect preimage of a metrizable space.(2)A topological group G is a strong q-space if and only if it is a strongly sequential-perfect preimage of a metrizable space.(3)A topological group G is a pointwise pseudocompact space if and only if there is a continuous open mapping f from G onto a metrizable space M such that f−1(F) is an r-pseudocompact set in G for each r-pseudocompact set F in M.Secondly, we study completeness of generalized countably compact topological groups in terms of preimages of completely metrizable spaces and obtain the following results:(4)A topological group G is strictly countably sieve complete if and only if G contains a closed sequentially compact subgroup H such that the quotient space G/H is completely metrizable and the canonical quotient mapping π:G→G/H is sequential-perfect.(5)A topological group G is countably sieve-s-complete if and only if G contains a closed sequentially compact subgroup H such that the quotient space G/H is completely metrizable and the canonical quotient mapping π:G→G/H is strongly sequential-perfect.(6)A topological group G is countably sieve-p-complete if and only if G contains a closed r-pseudocompact subgroup H such that the quotient space G/H is completely metrizable and the canonical quotient mapping π:G→G/H satisfies that π−1(F) is r-pseudocompact in G for each r-pseudocompact set F in G/H.

Projectively first-countable semitopological groups with certain D-properties

In this article, we give an internal characterization of subgroups of products of semitopological groups which satisfy certain properties that imply D-property. For example, we give an internal characterization of subgroups of products of regular semitopological groups which satisfy open (G), give an internal characterization of subgroups of products of regular first-countable semitopological groups which satisfy property (σ-A) (property (σ-B)). Every first-countable semitopological group with Collins-Roscoe property satisfies pre-(G) and property (pre-σ-B). We finally show that if G is a Hausdorff countably compact semitopological group with Hs(G)≤ω and G satisfies property (pre-σ-B) (pre-(G)), then G is a topological group.

Algebraic structure of countably compact non-torsion Abelian groups of size continuum from selective ultrafilters

Assuming the existence of c incomparable selective ultrafilters, we classify the non-torsion Abelian groups of cardinality c that admit a countably compact group topology. We show that for each κ∈[c,2c] each of these groups has a countably compact group topology of weight κ without non-trivial convergent sequences and another that has convergent sequences.Assuming the existence of 2c selective ultrafilters, there are at least 2c non homeomorphic such topologies in each case and we also show that every Abelian group of cardinality at most 2c is algebraically countably compact. We also show that it is consistent that every Abelian group of cardinality c that admits a countably compact group topology admits a countably compact group topology without non-trivial convergent sequences whose weight has countable cofinality.

On countably compact group topologies without non-trivial convergent sequences on [formula omitted] for arbitrarily large κ and a selective ultrafilter

Castro Pereira and Tomita showed that it is consistent that there exist arbitrarily large countably compact groups without non-trivial convergent sequences of finite order. In this paper, we obtain large p-compact groups without non-trivial convergent on the direct sum of κ copies of Q where κ=κω is infinite. In particular this gives the first arbitrarily large countably compact groups without non-trivial convergent sequences that are torsion-free.

Convergent sequences in topological groups

We survey recent developments concerning the role of convergent sequences in topological groups. We present the Invariant Ideal Axiom and announce its effect on convergence properties in topological groups, in particular, the consistency of the fact that every countable sequential topological group is either metrizable or kω. We also outline a construction of a countably compact topological group without non-trivial convergence sequences using iterated ultrapowers of Bohr topology on a Boolean group from a selective ultrafilter and announce a related ZFC construction of such a group. We recall the main open questions in the area and formulate several new ones.

On feebly compact paratopological groups

We obtain many results and solve some problems about feebly compact paratopological groups. We obtain necessary and sufficient conditions for such a group to be topological. One of them is the quasiregularity. We prove that each 2-pseudocompact paratopological group is feebly compact and that each Hausdorff σ-compact feebly compact paratopological group is a compact topological group. Our particular attention concerns periodic and topologically periodic groups. We construct examples of various compact-like paratopological groups which are not topological groups, among them a T0 sequentially compact group, a T1 2-pseudocompact group, a functionally Hausdorff countably compact group (under the axiomatic assumption that there is an infinite torsion-free Abelian countably compact topological group without non-trivial convergent sequences), and a functionally Hausdorff second countable group sequentially pracompact group. We prove that the product of a family of feebly compact paratopological groups is feebly compact, and that a paratopological group G is feebly compact provided it has a feebly compact normal subgroup H such that a quotient group G/H is feebly compact. For our research we also study some general constructions of paratopological groups. We extend the well-known construction of Raĭkov completion of a T0 topological group to the class of paratopological groups. We investigate cone topologies of paratopological groups which provide a general tool for constructing pathological examples, especially examples of compact-like paratopological groups with discontinuous inversion. We find a simple interplay between the algebraic properties of a basic cone subsemigroup S of a group G and compact-like properties of two basic semigroup topologies generated by S on the group G.

A countably compact topological group with the non-countably pracompact square

Under Martin's Axiom we construct a Boolean countably compact topological group whose square is not countably pracompact.

On some kinds of ω-balancedness in semitopological groups

In this article, we introduce two notions which are called M-ω-balancedness and C-ω-balancedness, respectively. We get the following conclusions. A regular semitopological group G is topologically isomorphic to a subgroup of the product of a family of Moore semitopological groups if and only if G is M-ω-balanced and Ir(G)≤ω. A T1 semitopological group G is topologically isomorphic to a subgroup of the product of a family of T1 semitopological groups with a point-countable base if and only if G is C-ω-balanced and Sm(G)≤ω. If G is a Hausdorff countably compact semitopological group with Hs(G)≤ω, then the M-ω-balancedness (C-ω-balancedness) of G implies that G is a topological group.Let G be an ω-balanced paratopological group and let e be the identity of G. We show that if for each U∈N(e) there exist ω-good set V∈N(e) with V⊂U and A⊂G such that ⋃{g(V∩V−1):g∈A}=G and {g(V∩V−1)V:g∈A} is star-countable, then G is completely ω-balanced, where N(e) is the family of open neighborhoods of e in G.

Countably compact groups and sequential order

We use ⋄ to construct, for every α≤ω1 a sequential countably compact topological group of sequential order α. This establishes the independence of the existence of sequential countably compact non Fréchet groups from the usual axioms of ZFC and answers several questions of D. Shakhmatov.

A van Douwen-like ZFC theorem for small powers of countably compact groups without non-trivial convergent sequences

We show that if κ≤ω and there exists a group topology without non-trivial convergent sequences on an Abelian group H such that Hn is countably compact for each n<κ then there exists a topological group G such that Gn is countably compact for each n<κ and Gκ is not countably compact. If in addition H is torsion, then the result above holds for κ=ω1. Combining with other results in the literature, we show that:a) Assuming c incomparable selective ultrafilters, for each n∈ω, there exists a group topology on the free Abelian group G such that Gn is countably compact and Gn+1 is not countably compact. (It was already know for ω).b) If κ∈ω∪{ω}∪{ω1}, there exists in ZFC a topological group G such that Gγ is countably compact for each cardinal γ<κ and Gκ is not countably compact.

No interesting sequential groups

We prove that it is consistent with ZFC that no sequential topological groups of intermediate sequential orders exist. This shows that the answer to a 1981 question of P. Nyikos is independent of the standard axioms of set theory. The model constructed also provides consistent answers to several questions of D. Shakhmatov, S. Todorčević and Uzcátegui. In particular, we show that it is consistent with ZFC that every countably compact sequential group is Fréchet–Urysohn.

The weight and Lindelöf property in spaces and topological groups

We show that if Y is a dense subspace of a Tychonoff space X, then w(X)≤nw(Y)Nag(Y), where Nag(Y) is the Nagami number of Y. In particular, if Y is a Lindelöf Σ-space, then w(X)≤nw(Y)ω≤nw(X)ω.Better upper bounds for the weight of topological groups are given. For example, if a topological group H contains a dense subgroup G such that G is a Lindelöf Σ-space, then w(H)=w(G)≤ψ(G)ω. Further, if a Lindelöf Σ-space X generates a dense subgroup of a topological group H, then w(H)≤2ψ(X).Several facts about subspaces of Hausdorff separable spaces are established. It is well known that the weight of a separable Hausdorff space X can be as big as 22c. We prove on the one hand that if a regular Lindelöf Σ-space Y is a subspace of a separable Hausdorff space, then w(Y)≤2ω, and the same conclusion holds for a Lindelöf P-space Y. On the other hand, we present an example of a countably compact topological Abelian group G which is homeomorphic to a subspace of a separable Hausdorff space and satisfies w(G)=22c, i.e. G has the maximal possible weight.

Density character of subgroups of topological groups

We give a complete characterization of subgroups of separable topological groups. Then we show that the following conditions are equivalent for an ω \omega -narrow topological group G G : (i) G G is homeomorphic to a subspace of a separable regular space; (ii) G G is topologically isomorphic to a subgroup of a separable topological group; (iii) G G is topologically isomorphic to a closed subgroup of a separable path-connected, locally path-connected topological group. A pro-Lie group is a projective limit of finite-dimensional Lie groups. We prove here that an almost connected pro-Lie group is separable if and only if its weight is not greater than the cardinality c \mathfrak {c} of the continuum. It is deduced from this that an almost connected pro-Lie group is separable if and only if it is homeomorphic to a subspace of a separable Hausdorff space. It is also proved that a locally compact (even feathered) topological group G G which is a subgroup of a separable Hausdorff topological group is separable, but the conclusion is false if it is assumed only that G G is homeomorphic to a subspace of a separable Tychonoff space. We show that every precompact (abelian) topological group of weight less than or equal to c \mathfrak {c} is topologically isomorphic to a closed subgroup of a separable pseudocompact (abelian) group of weight c \mathfrak {c} . This result implies that there is a wealth of closed non-separable subgroups of separable pseudocompact groups. An example is also presented under the Continuum Hypothesis of a separable countably compact abelian group which contains a non-separable closed subgroup.

A note on the continuity of the inverse in paratopological groups

The problem when a paratopolgical group (or semitopological group) is a topological group is interesting and important. In this paper, we continue to study this problem. It mainly shows that: (1) Let G be a paratopological group and put τ = ωHs(G); then G is a topological group if G is a Pτ-space; (2) every co-locally countably compact paratopological group G with ωHs(G) ≦ ω is a topological group; (3) every co-locally compact paratopological group is a topological group; (4) each 2-pseudocompact paratopological group G with ωHs(G) ≦ ω is a topological group. These results improve some results in [11, 13].

A countably compact free Abelian group of size continuum that admits a non-trivial convergent sequence

We show that it is consistent with ZFC that the free Abelian group of cardinality c admits a topological group topology that makes it countably compact with a non-trivial convergent sequence.

Abelian torsion groups with a countably compact group topology

Comfort and Remus [W.W. Comfort, D. Remus, Abelian torsion groups with a pseudocompact group topology, Forum Math. 6 (3) (1994) 323–337] characterized algebraically the Abelian torsion groups that admit a pseudocompact group topology using the Ulm–Kaplansky invariants. We show, under a condition weaker than the Generalized Continuum Hypothesis, that an Abelian torsion group (of any cardinality) admits a pseudocompact group topology if and only if it admits a countably compact group topology. Dikranjan and Tkachenko [D. Dikranjan, M. Tkachenko, Algebraic structure of small countably compact Abelian groups, Forum Math. 15 (6) (2003) 811–837], and Dikranjan and Shakhmatov [D. Dikranjan, D. Shakhmatov, Forcing hereditarily separable compact-like group topologies on Abelian groups, Topology Appl. 151 (1–3) (2005) 2–54] showed this equivalence for groups of cardinality not greater than 2 c . We also show, from the existence of a selective ultrafilter, that there are countably compact groups without non-trivial convergent sequences of cardinality κ ω , for any infinite cardinal κ. In particular, it is consistent that for every cardinal κ there are countably compact groups without non-trivial convergent sequences whose weight λ has countable cofinality and λ > κ .

Self-duality in the class of precompact groups

A topological Abelian group G is called ( strongly) self-dual if there exists a topological isomorphism Φ : G → G ∧ of G onto the dual group G ∧ (such that Φ ( x ) ( y ) = Φ ( y ) ( x ) for all x , y ∈ G ). We prove that every countably compact self-dual Abelian group is finite. It turns out, however, that for every infinite cardinal κ with κ ω = κ , there exists a pseudocompact, non-compact, strongly self-dual Boolean group of cardinality κ.

Countably Compact Paratopological Groups

By means of the theory of bispaces we show that a countably compact T0 paratopological group (G, τ) is a topological group if and only if (G, τ ∨ τ-1) is ω-bounded (here τ-1 is the conjugate topology of τ). Our approach is premised on the fact that every paratopological countably compact paratopological group is a Baire space and on the notion of a 2-pseudocompact space. We also prove that every ω-bounded (respectively, topologically periodic) Baire paratopological group admits a weaker Hausdorff group topology. In particular, ω-bounded (respectively, topologically periodic) 2-pseudocompact (so, also countably compact) paratopological groups enjoy this property. Some topological properties turning countably compact topological semigroups into topological groups are presented and some open questions are posed.

Countably compact topological group topologies on free Abelian groups from selective ultrafilters

Tkachenko showed in 1990 the existence of a countably compact group topology on the free Abelian group of size c using CH. Koszmider, Tomita and Watson showed in 2000 the existence of a countably compact group topology on the free Abelian group of size 2 c using a forcing model in which CH holds. Wallace's question from 1955, asks whether every both-sided cancellative countably compact semigroup is a topological group. A counterexample to Wallace's question has been called a Wallace semigroup. In 1996, Robbie and Svetlichny constructed a Wallace semigroup under CH. In the same year, Tomita constructed a Wallace semigroup from MA countable . In this note, we show that the examples of Tkachenko, Robbie and Svetlichny, and Koszmider, Tomita and Watson can be obtained using a family of selective ultrafilters. As a corollary, the constructions presented here are compatible with the total failure of Martin's Axiom.