The Ellis theorem and continuity in 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 answer (Arhangel’skii et al., Math Maced 8:1–19, 2010, Problem 10.4), by showing that if (G, ·, τ) is a semitopological group and (G, τ) is homeomorphic to a product of Cech-complete spaces, then (G, ·, τ) is a topological group.

In this note, we firstly discuss some properties of spaces which are countably sieve-complete, densely q-complete and strongly Baire. By some known conclusions, we finally show that if G is a regular countably sieve-complete semitopological group then G is a topological group. If a regular semitopological group G has a dense subgroup which is countably sieve-complete (densely q-complete), then G is a topological group. If G is a regular countably sieve-complete semitopological group then G is a D-space if and only if G is paracompact. We point out that some conditions in Theorem 2.14 and Corollary 2.15 in [17] are not essential.

LetS be a uniform semigroup (this includes all topological groups and affine semigroups). We show that a certain space of uniformly continuous functions onS has a left invariant mean iffS has the fixed point property for uniformly continuous affine actions ofS on compact convex sets. This is closely related to but independent of the results of T. Mitchell in [13] and A. Lau in [10]. Interesting examples and consequences are given for the special cases of topological groups and affine convolution semigroups of probability measures on a locally compact semigroup or group.

In this paper, it is shown that every compact Hausdorff $K$-space has countable tightness. This result gives a positive answer to a problem posed by Malykhin and Tironi [‘Weakly Fréchet–Urysohn and Pytkeev spaces’, Topology Appl.104 (2000), 181–190]. We show that a semitopological group $G$ that is a $K$-space is first countable if and only if $G$ is of point-countable type. It is proved that if a topological group $G$ is a $K$-space and has a locally paracompact remainder in some Hausdorff compactification, then $G$ is metrisable.

M. Sion and T. Traynor investigated ([15]-[17]), measures and integrals having values in topological groups or semigroups. Their definition of integrability was a modification of Phillips-Rickart bilinear vector integrals, in locally convex topological vector spaces.The purpose of this paper is to develop a good notion of an integration process in partially ordered groups, based on their order structure. The results obtained generalize some of the results of J. D. M. Wright ([19]-[22]) where the measurable functions are real-valued and the measures take values in partially ordered vector spaces.Let if be a σ-algebra of subsets of T, X a lattice group, Y, Z partially ordered groups and m : H → F a F-valued measure on H. By F(T, X), M(T, X), E(T, X) and S(T, X) are denoted the lattice group of functions with domain T and with range X, the lattice group of (H, m)-measurable functions of F(T, X) and the lattice group of (H, m)-elementary measurable functions of F(T, X) and the lattice group of (H, m)-simple measurable functions of F(T, X) respectively.

Totally Lindelöf and totally ω-narrow paratopological groups

Topological groups with thin generating sets

ABSTRACT. Some results of Mukherjea and Tserpes are generalized by showing that any sequentially compact, cancellative topological semigroup is a topological group. Hence, any countably compact, cancellative topological semigroup with any additional condition that would imply sequential compactness is also a topological group. Finally, it is shown that any w‐bounded, cancellative topological semigroup is also a topological group.

Metrizability of topological groups having weak topologies with respect to good covers

Countably compact sequential topological semigroup with two-sides cancellations is a topological group

We prove that the existence of a selective ultrafilter implies the existence of a countably compact Hausdorff group topology on the free Abelian group of size continuum. As a consequence, we show that the existence of a selective ultrafilter implies the existence of a Wallace semigroup (i.e., a countably compact both-sided cancellative topological semigroup which is not a topological group).

A classification theorem for diagrams of simplicial sets

Some conditions equivalent to a strong quasi‐divisor property (SQDP) for a partly ordered group G are derived. It is proved that if G is defined by a family of t‐valuations of finite character, then G admits an SQDP if and only if it admits a quasi‐divisor property and any finitely generated t‐ideal is generated by two elements. A topological density condition in topological group of finitely generated t‐ideals and/or compatible elements are proved to be equivalent to SQDP.

ABSTRACTS OF PAPERS TO APPEAR IN FUTURE ISSUES

Topological Field Theory and Quantum Holonomy Representations of Motion Groups