Generalized metric spaces with algebraic structures

We discuss generalized metrizable properties on paratopological groups and topological groups. It is proved in this paper that a first-countable paratopological group which is a β-space is developable; and we construct a Hausdorff, separable, non-metrizable paratopological group which is developable. We consider paratopological (topological) groups determined by a point-countable first-countable subspaces and give partial answers to Arhangel'skii's conjecture; Nogura-Shakhmatov-Tanaka's question (Nogura et al., 1993 [23]). We also give a negative answer to a question in Cao et al. (in press) [10]. Finally, remainders of topological groups and paratopological groups are discussed and Arhangel'skii's Theorem (Arhangel'skii, 2007 [3]) is improved.     

Even continuity and topological equicontinuity in topologized semigroups

A topologized semigroup X having an evenly continuous resp., topologically equicontinuous, family RX of right translations is investigated. It is shown that: (1) every left semitopological semigroup X with an evenly continuous family RX is a topological semigroup, (2) a semitopological group X is a paratopological group if and only if the family RX is evenly continuous and (3) a semitopological group X is a topological group if and only if the family RX is topologically equicontinuous. In particular, we get that for any paratopological group X which is not a topological group, the family RX provides an example of a transitive group of homeomorphisms of X that is evenly continuous and not topologically equicontinuous. The last conclusion answers negatively a question posed by H.L. Royden.     

Notes on paratopological groups

In this paper, we prove that if a remainder of a non-locally compact paratopological group G   has a _Gδ$G_{\delta }$-diagonal and every compact subset of G is first countable, then G   has a _Gδ$G_{\delta }$-diagonal of infinite rank. This improves a result of Chuan Liu and Shou Lin [Chuan Liu, Shou Lin, Generalized metric spaces with algebraic structure, Topology Appl. 157 (2010) 1966–1974]. We also construct an open continuous homomorphism f from a non-metrizable paratopological group G onto a metrizable topological group H such that the kernel of f is metrizable. This result gives a negative answer to an open problem posed in [A.V. Arhangel?skii, M. Tkachenko, Topological Groups and Related Structures, Atlantis Press, World Scientific, 2008].     

On an affirmative answer to S. Lin?s problem

In this paper, we give an affirmative answer to the problem posed by S. Lin (2002, 2007) in [7] and [8], and give another answer to the question posed by Y. Ikeda, C. Liu and Y. Tanaka (2002) in [5].     

Semitopological groups,Bouziad spaces and topological groups

A semitopological group (topological group) is a group endowed with a topology for which multiplication is separately continuous (multiplication is jointly continuous and inversion is continuous). In this paper we use topological games to show that many semitopological groups are in fact topological groups.     

Fréchet-Urysohn fans in free topological groups

In this paper we answer the question of T. Banakh and M. Zarichnyi constructing a copy of the Fréchet-Urysohn fan S_ω in a topological group G admitting a functorial embedding [0,1]⊂G. The latter means that each autohomeomorphism of [0,1] extends to a continuous homomorphism of G. This implies that many natural free topological group constructions (e.g. the constructions of the Markov free topological group, free abelian topological group, free totally bounded group, free compact group) applied to a Tychonov space X containing a topological copy of the space Q of rationals give topological groups containing S_ω.     

Observations about equicontinuity and related concepts

The concepts of equicontinuity, even continuity, topological equicontinuity and the newly defined concepts of compact equicontinuity and compact topological equicontinuity are compared. It is shown that for a set of group homomorphisms from a semitopological group to a topological group all these notions of equicontinuities coincide. It is shown also that an infinite-dimensional normed space endowed with its weak topology is an example of a space which does not satisfy the Ascoli theorem in Noble's sense.     

On weak bases

In this paper, we give an affirmative answer to Tanaka's question: Is a space X with a σ-hereditarily closure-preserving weak base g-metrizable? [Proc. Aroc. Amer. Math. Soc. 112 (1991) 283] and a negative answer to S. Lin's question: Is every weak base of a topological space a k-network? [S. Lin, Generalized Metric Spaces and Maps, Science Press, 1995, Problem 1.6.20]. We also discuss mapping theorems on weak bases and the product of weak bases.     

Embedding paratopological groups into topological products

We show that a Hausdorff paratopological group G admits a topological embedding as a subgroup into a topological product of Hausdorff first-countable (second-countable) paratopological groups if and only if G is ω-balanced (totally ω-narrow) and the Hausdorff number of G is countable, i.e., for every neighbourhood U of the neutral element e of G there exists a countable family γ of neighbourhoods of e such that _?V∈γVV−1⊆U. Similarly, we prove that a regular paratopological group G can be topologically embedded as a subgroup into a topological product of regular first-countable (second-countable) paratopological groups if and only if G is ω-balanced (totally ω-narrow) and the index of regularity of G is countable.As a by-product, we show that a regular totally ω-narrow paratopological group with countable index of regularity is Tychonoff.     

On metric spaces and local extrema

The main aim of this paper is to give a positive answer to a question of Behrends, Geschke and Natkaniec regarding the existence of a connected metric space and a non-constant real-valued continuous function on it for which every point is a local extremum. Moreover we show that real-valued continuous functions on connected spaces such that every family of pairwise disjoint non-empty open sets is of size <|R| are constant provided that every point is a local extremum.     

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.     

More on remainders close to metrizable spaces

This article is a natural continuation of [A.V. Arhangel'skii, Remainders in compactifications and generalized metrizability properties, Topology Appl. 150 (2005) 79-90]. As in [A.V. Arhangel'skii, Remainders in compactifications and generalized metrizability properties, Topology Appl. 150 (2005) 79-90], we consider the following general question: when does a Tychonoff space X have a Hausdorff compactification with a remainder belonging to a given class of spaces? A famous classical result in this direction is the well known theorem of M. Henriksen and J. Isbell [M. Henriksen, J.R. Isbell, Some properties of compactifications, Duke Math. J. 25 (1958) 83-106].It is shown that if a non-locally compact topological group G has a compactification bG such that the remainder Y=bG?G has a Gδ-diagonal, then both G and Y are separable and metrizable spaces (Theorem 5). Several corollaries are derived from this result, in particular, this one: If a compact Hausdorff space X is first countable at least at one point, and X can be represented as the union of two complementary dense subspaces Y and Z, each of which is homeomorphic to a topological group (not necessarily the same), then X is separable and metrizable (Theorem 12). It is observed that Theorem 5 does not extend to arbitrary paratopological groups. We also establish that if a topological group G has a remainder with a point-countable base, then either G is locally compact, or G is separable and metrizable.     

On countable-to-one maps

In this paper, it is proved that a space with a point-countable base is an open, countable-to-one image of a metric space, and a quotient, countable-to-one image of a metric space is characterized by a point-countable ℵ₀-weak base. Examples are provided in order to answer negatively questions posed by Gruenhage et al. [G. Gruenhage, E. Michael, Y. Tanaka, Spaces determined by point-countable covers, Pacific J. Math. 113 (1984) 303-332] and Tanaka [Y. Tanaka, Closed maps and symmetric spaces, Questions Answers Gen. Topology 11 (1993) 215-233].     

Reflexive representability and stable metrics

It is well-known that a topological group can be represented as a group of isometries of a reflexive Banach space if and only if its topology is induced by weakly almost periodic functions [see Megrelishvili (Operator topologies and reflexive representability. Nuclear groups and Lie groups (Madrid 1999). Res. Exp. Math., vol. 24, pp. 197?C208. Heldermann, Lemgo, 2001; Topological transformation groups: selected topics. In: Pearl E (ed) Open Problems in Topology II. Elsevier, Amsterdam 2007), Shtern (Russian J. Math. Phys. 2(1):131?C132, 1994)]. We show that for a metrisable group this is equivalent to the property that its metric is uniformly equivalent to a stable metric in the sense of Krivine and Maurey (Isr J Math 39(4):273?C295, 1981). This result is used to give a partial negative answer to a problem of Megrelishvili.     

Remainders of rectifiable spaces

We prove a Dichotomy Theorem: for any Hausdorff compactification bG of an arbitrary rectifiable space G, the remainder bG?G is either pseudocompact or Lindelöf. This theorem generalizes a similar theorem on topological groups obtained earlier in A.V. Arhangel'skii (2008) [6], but the proof for rectifiable spaces is considerably more involved than in the case of topological groups. It follows that if a remainder of a rectifiable space G is paracompact or Dieudonné complete, then the remainder is Lindelöf and that G is a p-space. We also present an example showing that the Dichotomy Theorem does not extend to all paratopological groups. Some other results are obtained, and some open questions are formulated.     

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.     

Dense orbits of automorphisms and compactness of groups

It is proved, by using topological properties, that when a group automorphism of a locally compact totally disconnected group is ergodic under the Haar measure, the group is compact. The result is an answer for Halmos's question that has remained open for the totally disconnected case.     

Topological games and continuity of group operations

We consider a topological game GΠ involving two players α and β and show that, for a paratopological group, the absence of a winning strategy for player β implies the group is a topological one. We provide a large class of topological spaces X for which the absence of a winning strategy for player β is equivalent to the requirement that X is a Baire space. This allows to extend the class of paratopological or semitopological groups for which one can prove that they are, actually, topological groups.Conditions of the type “existence of a winning strategy for the player α” or “absence of a winning strategy for the player β” are frequently used in mathematics. Though convenient and satisfactory for theoretical considerations, such conditions do not reveal much about the internal structure of the topological space where they hold. We show that the existence of a winning strategy for any of the players in all games of Banach-Mazur type can be expressed in terms of “saturated sieves” of open sets.     

The rational line partitions every self-dense metrisable space

In the present note we shall prove that a metrisable space can be partitioned by the rational line if that space is self-dense. This gives an affirmative answer to a question raised by Rankston-McGovern [1].     

When is a Volterra space Baire?

In this paper, we study the problem when a Volterra space is Baire. It is shown that every stratifiable Volterra space is Baire. This answers affirmatively a question of Gruenhage and Lutzer in [G. Gruenhage, D. Lutzer, Baire and Volterra spaces, Proc. Amer. Math. Soc. 128 (2000) 3115-3124]. Further, it is established that a locally convex topological vector space is Volterra if and only if it is Baire; and the weak topology of a topological vector space fails to be Baire if the dual of the space contains an infinite linearly independent pointwise bounded subset.