Paratopological Groups Research Articles

Cross-complementary subsets in paratopological groups

The following general question is considered by A.V. Arhangel’skiˇi [Perfect mappings in topological groups, cross-complementary subsets and quotients, Comment. Math.Univ.Carolin., 2003]. Suppose that G is a topological group, and F,M are subspaces of G such that G = MF. Under these general assumptions, how are the properties of F andM related to the properties of G? Also, A.V. Arhangel’skiˇi and M. Tkachenko [Topological Groups and Related Structures, Atlantis Press, World Sci., 2008] asked what is about the above question in paratopological groups [Open problem 4.6.9, Topological Groups and Related Structures, Atlantis Press, World Sci., 2008]. In this paper, we mainly consider this question and some positive answers to this question are given. In particular, we find many A.V. Arhangel’skiˇi’s results hold for k-gentle paratopological groups.

Subgroups of products of certain paratopological (semitopological) groups

In the first part of this note, we give some sufficient conditions under which a paratopological group is topologically isomorphic to a subgroup of a product of strongly metrizable paratopological groups. In the second part of this note, we show that a regular (Hausdorff, T1) semitopological group G admits a homeomorphic embedding as a subgroup into a product of regular (Hausdorff, T1) first-countable semitopological groups which are σ-spaces if and only if G is locally ω-good, ω-balanced, Ir(G)≤ω (Hs(G)≤ω, Sm(G)≤ω) and with the property that for every open neighborhood U of the identity e of G the cover {xU:x∈G} has a basic refinement F which is σ-discrete with respect to a countable family V of open neighborhoods of e. In the last part of this note, we give an internal characterization of projectively Ti second-countable semitopological groups, for i=0,1,2.

Quasi-boundedness of irresolute paratopological groups

Continuing the study of irresolute paratopological groups, our focus in this paper is to define and study the boundedness for irresolute paratopological groups. Premeager property for irresolute pa...

The regularity of quotient paratopological groups

Let H be a closed subgroup of a regular abelian paratopological group G. The group reflexion G ( of G is the group G endowed with the strongest group topology, weaker that the original topology of G. We show that the quotient G/H is Hausdorff (and regular) if H is closed (and locally compact) in G ( . On the other hand, we construct an example of a regular abelian paratopological group G containing a closed discrete subgroup H such that the quotient G/H is Hausdorff but not regular. In this paper we study the properties of the quotients of paratopological groups by their normal subgroups. By a paratopological group G we understand a group G endowed with a topology � making the group operation continuous, see (ST). If, in addition, the operation of taking inverse is continuous, then the paratopological group (G;�) is a topological group. A standard example of a paratopological group failing to be a topological group is the Sorgefrey line L, that is the real line R endowed with the Sorgefrey topology (generated by the base consisting of half-intervals (a;b), a < b). Let (G;�) be a paratopological group and HG be a closed normal subgroup of G. Then the quotient group G=H endowed with the quotient topology is a paratopological group, see (Ra). Like in the case of topological groups, the quotient homomorphism � : G ! G=H is open. If the subgroup HG is compact, then the quotient G=H is Hausdorff (and regular) provided so is the group G, see (Ra). The compactness of H in this result cannot be replaced by the local compactness as the following simple example shows. Example 1. The subgroup H = f(−x;x) : x 2 Qg is closed and discrete in the square G = L 2 of the Sorgenfrey line L. Nonetheless, the quotient group G=H fails to be Hausdorff: for any irrational x the coset (−x;x) + H cannot be separated from zero (0;0) + H. A necessary and sufficient condition for the quotientG=H to be Hausdorff is the closedness of H in the topology of group reflexion G ( of G. By the group reflexion G ( = (G;� ( ) of a paratopological group (G;�) we under- stand the group G endowed with the strongest topology � ( � � turning G into a topological group. This topology admits a categorial description: � ( is a unique topology on G such that � (G;� ( ) is a topological group; � the identity homomorphism id : (G;�) ! (G;� ( ) is continuous;

O-Tightness in paratopological groups

In this paper, we mainly discuss the o-tightness in paratopological groups. The following results are obtained: (1) Every paratopological group H satisfying Sm(H)⋅get(H)≤ω is Gδ-preserving. (2) The o-tightness of the product space X×H is countable for every Hausdorff feathered paratopological group H and every space X with ot(X)≤ω. As an application, we deduce that every Hausdorff feathered paratopological group H has countable o-tightness; in particular, H is a Moscow space, which gives a positive answer to [2, Open problem 6.4.8].

Characterizing s-paratopological groups by free paratopological groups

A paratopological group G is called an s-paratopological group if every sequentially continuous homomorphism from G to a paratopological group is continuous. In this paper, the structure of s-paratopological groups is established in terms of free paratopological groups. Namely, if G is a non-discrete T1 paratopological group, then the following statements (1), (2), (3) and (4) are equivalent. (1) G is an s-paratopological group. (2) G is topologically isomorphic to a quotient group of a free paratopological group on a metrizable space. (3) G is topologically isomorphic to a quotient group of a free paratopological group on a T1 Fréchet space. (4) G is topologically isomorphic to a quotient group of a free paratopological group on a T1 sequential space.

Remainders for a class of quotients of paratopological groups

The metrizability conditions and cardinal invariants for compactly-fibered coset spaces in paratopological groups in terms of their remainders are considered. It mainly shows that: (1) Suppose that X is a quotient of a k-gentle paratopological group with respect to a compact subgroup; then the remainder Y of X is either pseudocompact or Lindelöf; (2) Suppose that X is a quotient of paratopological group with respect to a compact subgroup with a remainder Y such that Y has a point-countable base; then X and Y are separable and metrizable; (3) Suppose that X is a quotient of paratopological group with respect to a compact subgroup with a remainder Y such that Y is weakly developable; then nw(X)=πw(X)=πw(Y)=ω. Many results on coset spaces of topological groups obtained by A.V. Arhangel'skiı̌ [5] are extended to coset spaces of paratopological groups.

Partially paratopological groups

In this paper, we introduce the notion of a partially (para)topological group, using generalized topology in the sense of H. Delfs and M. Knebusch. We study some basic properties of this kind of structures. Among them, we prove that an important theorem of Banakh and Ravsky, applied to their proof in ZFC that every regular paratopological group is completely regular, can be false in a model for ZF. Therefore, it is an open problem whether there exists a model for ZF in which a Hausdorff topological group need not be completely regular. We notice that Banakh–Ravsky theorem that every regular paratopological group is completely regular is provable in every model for ZF+DC.

C-compactness in locally compact groups and paratopological groups

We study c-compactness in two cases. First, we obtain some subclasses of locally compact groups where compactness and c-compactness coincide, and besides, a result due to Dikranjan and Uspenskij is generalized. We introduce c-compactness and h-completeness in the wider class of Hausforff paratopological groups. It is proved that the closure of any subgroup of a c-compact paratopological group is again a subgroup. We present an example of a non-compact h-complete paratopological group.

A non-ω-balanced paratopological group with countable π-character

In this paper, we mainly prove that if G is a saturated paratopological group with we(G)≤κ, where κ is an infinite cardinal, then G is κ-narrow. It gives a partial answer to the problem posed by Sánchez in [10, Problem 2.24] and even more. Applying this property, we show that if G is a saturated Hausdorff paratopological group with we(G)πχ(G)≤ω, then G can be condensed onto a Hausdorff space with a countable base. Also, we construct a Hausdorff paratopological group with countable π-character, but it is not ω-balanced. This gives a negative answer to the problem posed by Sánchez in [9, Problem 2.13].

The countable type properties in free paratopological groups

A space X is of countable type (resp. subcountable type) if every compact subspace F of X is contained in a compact subspace K that is of countable character (resp. countable pseudocharacter) in X. In this paper, we mainly show that: (1) For a functionally Hausdorff space X, the free paratopological group FP(X)and the free abelian paratopological group AP(X) are of countable type if and only if X is discrete; (2) For a functionally Hausdorff space X, if the free abelian paratopological group AP(X) is of subcountable type then X has countable pseudocharacter. Moreover, we also show that, for an arbitrary Hausdorff μ-space X, if AP2(X) or FP2(X) is locally compact, then X is a topological sum of a compact space and a discrete space.

Spaces which are retracts or cofactors of paratopological groups

In this paper we investigate Tychonoff spaces which are retracts of paratopological groups. A strong necessary condition for that is the existence of a certain binary operation on the space (called a τ-twister), which was introduced in [2,3]. Some general theorems are established which imply that βω is not a retract of a paratopological group. We also notice, using some deep results of V.V. Uspenskij, that the space ω1 of countable ordinals is not a retract of any topological group (see Fact 3).

Quasi-uniformities and quotients of paratopological groups

If $H$ is a subgroup of a paratopological group $G$ , we prove that the quotient topology of the coset space $G/H$ is induced by a point-rotund uniformity and the quotient topology of the semiregularization $(G/H)_{sr}$ of $G/H$ is induced by a normal quasi-uniformity. In particular, $(G/H)_{sr}$ is a Tychonoff space provided that $G/H$ is Hausdorff . The previous results are applied in order to show that every Hausdorff Lindel o f paratopological group is $\omega$ -admissible. We also show that, if $G$ is an $\omega$ -admissible paratopological group, then so is the reflections $T_i(G)$ (i=1,2,3), $Reg(G)$ and $Tych(G)$ .

Mapping i2 on the free paratopological groups

Let FP(X) be the free paratopological group over a topological space X. For each nonnegative integer n ? N, denote by FPn(X) the subset of FP(X) consisting of all words of reduced length at most n, and in by the natural mapping from (X ? X?1 ? {e})n to FPn(X). We prove that the natural mapping i2:(X ? X?1 d ?{e})2 ? FP2(X) is a closed mapping if and only if every neighborhood U of the diagonal ?1 in Xd x X is a member of the finest quasi-uniformity on X, where X is a T1-space and Xd denotes X when equipped with the discrete topology in place of its given topology.

On reflections and three space properties of semi(para)topological groups

We show that many topological properties are invariant and/or inverse invariant under taking T2-reflections in semitopological groups. We also extend some three space properties in topological groups (paratopological groups) to semitopological groups. The following result is established: Let G be a regular semitopological group and let H be a closed subgroup of G such that all compact (resp., countably compact, sequentially compact) subsets of the semitopological group H are first-countable. If the quotient space G/H has the following property, then so does the semitopological group G.(*) all compact (resp., countably compact, sequentially compact) subsets are Hausdorff and strongly Fréchet (strictly Fréchet).

Three-space properties in paratopological groups

We show that neither first-countable nor second-countable are three-space properties in the class of paratopological groups: we present a countable regular paratopological Abelian group H which contains a closed discrete subgroup F such that H/F is topologically isomorphic to the rational numbers with the Sorgenfrey topology and H is not first-countable. Also, we prove that if H is an invariant topological subgroup of a paratopological group G such that H is second-countable and G/H has countable network, then G has countable network as well (this answers a question posed in [12]). Hence if H is an invariant topological subgroup of a first-countable paratopological group G such that H and G/H are second-countable, then so is G.

MP-equivalence of free paratopological groups

Let FP(X) denote the free paratopological group over a topological space X. Two topological spaces X and Y are called MP-equivalent if FP(X) and FP(Y) are topologically isomorphic. At first, it is shown that there exist non-homeomorphic topological spaces X and Y such that FP(X) and FP(Y) are topologically isomorphic. Secondly, MP-invariance of free paratopological groups is investigated. It is established that pseudocompactness, hereditary Lindelöfness, hereditary separability and the property of being a cosmic space are all MP-invariant, which generalizes some conclusions valid for free topological groups to free paratopological groups. Finally, a few questions about MP-equivalence of free paratopological groups are posed.

Each regular paratopological group is completely regular

We prove that a semiregular topological space X X is completely regular if and only if its topology is generated by a normal quasi-uniformity. This characterization implies that each regular paratopological group is completely regular. This resolves an old problem in the theory of paratopological groups, which stood open for about 60 years. Also we define a natural uniformity on each paratopological group and using this uniformity prove that each (first countable) Hausdorff paratopological group is functionally Hausdorff (and submetrizable). This resolves another two known open problems in the theory of paratopological groups.

Productivity of Coreflective Subcategories of Semitopological Groups

The present paper generalizes to semitopological and quasitopological groups some results achieved by Horst Herrlich and the second author for topological groups. The results concern preserving products in coreflective subcategories. Unlike in paratopological or topological groups, there are non-finitely productive bicoreflective subcategories of quasitopological groups. We desctribe bicoreflective subcategories of semitopological groups that are either finitely productive or productive or their productivity number is submeasurable. To achieve the results, several general factorization theorems for maps on products will be proved with the help of modification of NSS property of groups.

Completions of paratopological groups and bounded sets

In this paper we consider two questions in the realm of paratopological groups: When does multiplication on a given Tychonoff paratopological group H admit an extension to continuous multiplication on the Dieudonne completion, \(\mu {H}\), of H in such a way that H turns into a dense subgroup of the paratopological group \(\mu {H}\)? and, if A and B are bounded subsets of paratopological groups G and H, respectively, is \(A\times B\) bounded in \(G\times H\)? The motivation for these questions comes from the field of topological groups and they are receiving a special attention as an interesting source of open problems.