Invariant measures for equicontinuous semigroups of continuous transformations of a compact Hausdorff space

Equicontinuous semigroups of transformations of a compact Hausdorff space and their sets of all invariant (Borel, regular and probabilistic) measures are studied. Conditions equivalent to the existence of at least one invariant measure are given. The (algebraic and topological) structure of the set of invariant measures is researched.     

On the structure of finite coverings of compact connected groups

Finite-sheeted covering mappings onto compact connected groups are studied. We show that for a covering mapping from a connected Hausdorff topological space onto a compact (in general, non-abelian) group there exists a topological group structure on the covering space such that the mapping becomes a homomorphism of groups. To prove this fact we construct an inverse system of covering mappings onto Lie groups which approximates the given covering mapping. As an application, it is shown that a covering mapping onto a compact connected abelian group G must be a homeomorphism provided that the character group of G admits division by degree of the mapping. We also get a criterion for triviality of coverings in terms of means and prove that each finite covering of G is equivalent to a polynomial covering.     

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.     

Connected generalised Sierpiński carpets

Generalised Sierpiński carpets are planar sets that generalise the well-known Sierpiński carpet and are defined by means of sequences of patterns. We present necessary and sufficient conditions, under which generalised Sierpiński carpets are connected, with respect to Euclidean topology.     

Unconditionally τ-closed and τ-algebraic sets in groups

Families of unconditionally τ-closed and τ-algebraic sets in a group are defined, which are natural generalizations of unconditionally closed and algebraic sets defined by Markov. A sufficient condition for the coincidence of these families is found. In particular, it is proved that these families coincide in any group of cardinality at most τ. This result generalizes both Markov's theorem on the coincidence of unconditionally closed and algebraic sets in a countable group (as is known, they may be different in an uncountable group) and Podewski's theorem on the topologizability of any ungebunden group.     

Generalized selection theorems without convexity

In this paper, we extend new selection theorems for almost lower semicontinuous multifunctions T on a paracompact topological space X to general nonconvex settings. On the basis of the Kim-Lee theorem and the Horvath selection theorem, we first show that any a.l.s.c. C-valued multifunction admits a continuous selection under a mild condition of a one-point extension property. Finally, we apply a fundamental selection theorem, due to Ben-El-Mechaiekh and Oudadess, to modify our selection theorems by adjusting a closed subset Z of X with its covering dimension dim_XZ≤0. The results derived here generalize and unify various earlier ones from classic continuous selection theory.     

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].     

Almost maximally almost-periodic group topologies determined by T-sequences

A sequence {an} in a group G is a T-sequence if there is a Hausdorff group topology τ on G such that . In this paper, we provide several sufficient conditions for a sequence in an abelian group to be a T-sequence, and investigate special sequences in the Prüfer groups Z(p^∞). We show that for p≠2, there is a Hausdorff group topology τ on Z(p^∞) that is determined by a T-sequence, which is close to being maximally almost-periodic—in other words, the von Neumann radical n(Z(p^∞),τ) is a non-trivial finite subgroup. In particular, n(n(Z(p^∞),τ))?n(Z(p^∞),τ). We also prove that the direct sum of any infinite family of finite abelian groups admits a group topology determined by a T-sequence with non-trivial finite von Neumann radical.     

Locally compact abelian groups admitting non-trivial quasi-convex null sequences

In this paper, we show that, for every locally compact abelian group G, the following statements are equivalent:

(i)

G contains no sequence such that {0}∪{±x_n∣n∈N} is infinite and quasi-convex in G, and x_n?0;

(ii)

one of the subgroups {g∈G∣2g=0} or {g∈G∣3g=0} is open in G;

(iii)

G contains an open compact subgroup of the form or for some cardinal κ.

     

Group-valued continuous functions with the topology of pointwise convergence

Let G be a topological group with the identity element e. Given a space X, we denote by Cp(X,G) the group of all continuous functions from X to G endowed with the topology of pointwise convergence, and we say that X is: (a) G-regular if, for each closed set F⊆X and every point x∈X?F, there exist f∈Cp(X,G) and g∈G?{e} such that f(x)=g and f(F)⊆{e}; (b) G^?-regular provided that there exists g∈G?{e} such that, for each closed set F⊆X and every point x∈X?F, one can find f∈Cp(X,G) with f(x)=g and f(F)⊆{e}. Spaces X and Y are G-equivalent provided that the topological groups Cp(X,G) and Cp(Y,G) are topologically isomorphic.We investigate which topological properties are preserved by G-equivalence, with a special emphasis being placed on characterizing topological properties of X in terms of those of Cp(X,G). Since R-equivalence coincides with l-equivalence, this line of research “includes” major topics of the classical Cp-theory of Arhangel'ski? as a particular case (when G=R).We introduce a new class of TAP groups that contains all groups having no small subgroups (NSS groups). We prove that: (i) for a given NSS group G, a G-regular space X is pseudocompact if and only if Cp(X,G) is TAP, and (ii) for a metrizable NSS group G, a G^?-regular space X is compact if and only if Cp(X,G) is a TAP group of countable tightness. In particular, a Tychonoff space X is pseudocompact (compact) if and only if Cp(X,R) is a TAP group (of countable tightness). Demonstrating the limits of the result in (i), we give an example of a precompact TAP group G and a G-regular countably compact space X such that Cp(X,G) is not TAP.We show that Tychonoff spaces X and Y are T-equivalent if and only if their free precompact Abelian groups are topologically isomorphic, where T stays for the quotient group R/Z. As a corollary, we obtain that T-equivalence implies G-equivalence for every Abelian precompact group G. We establish that T-equivalence preserves the following topological properties: compactness, pseudocompactness, σ-compactness, the property of being a Lindelöf Σ-space, the property of being a compact metrizable space, the (finite) number of connected components, connectedness, total disconnectedness. An example of R-equivalent (that is, l-equivalent) spaces that are not T-equivalent is constructed.     

Compact groups containing dense pseudocompact subgroups without non-trivial convergent sequences

Let G be compact abelian group such that w(C(G))=w(Cω(G)). We prove that if

|C(G)|?m(G/C(G)),     

Automatic continuity of biseparating homomorphisms defined between groups of continuous functions

Let C(X,T) be the group of continuous functions of a compact Hausdorff space X to the unit circle of the complex plane T with the pointwise multiplication as the composition law. We investigate how the structure of C(X,T) determines the topology of X. In particular, which group isomorphisms H between the groups C(X,T) and C(Y,T) imply the existence of a continuous map h of Y into X such that H is canonically represented by h. Among other results, it is proved that C(X,T) determines X module a biseparating group isomorphism and, when X is first countable, the automatic continuity and representation as Banach-Stone maps for biseparating group isomorphisms is also obtained.     

Classes defined by stars and neighbourhood assignments

We apply and develop an idea of E. van Douwen used to define D-spaces. Given a topological property P, the class P^∗ dual to P (with respect to neighbourhood assignments) consists of spaces X such that for any neighbourhood assignment there is Y⊂X with Y∈P and . We prove that the classes of compact, countably compact and pseudocompact are self-dual with respect to neighbourhood assignments. It is also established that all spaces dual to hereditarily Lindelöf spaces are Lindelöf. In the second part of this paper we study some non-trivial classes of pseudocompact spaces defined in an analogous way using stars of open covers instead of neighbourhood assignments.     

Generalized Sierpinski sets

The notion of a Sierpinski topological space is introduced and some properties of such spaces connected with Borel measures are considered.     

Combinations not tolerated by normal topological groups

We show that if X is not paracompact then one can find a compact space K such that X⊕K does not embed closedly into any normal topological group.     

Scale function vs topological entropy

In the realm of topological automorphisms of totally disconnected locally compact groups, the scale function introduced by Willis in [19] is compared with the topological entropy. We prove that the logarithm of the scale function is always dominated by the topological entropy and we provide examples showing that this inequality can be strict. Moreover, we give a condition equivalent to the equality between these two invariants. Various properties of the scale function, inspired by those of the topological entropy, are presented.