Two Notes on Topological Groups

In this short paper, we firstly give a short proof of Birkhoff-Kakutani Theorem by Moore metrizable Theorem. Then we prove that G is a topological group if it is a paratopological group which is a dense G_δ-set in a locally feebly compact regular space X.

About Compact and Sequentially Compact Coincide in Topological Groups with the Axiom (A_1)

This paper is concerned with the relation between the compactness and sequential compactness in a topological space or a topological group, and show that the compactness and sequential compactness coincide in a topolpgical group with the axiom (A1).

THE MAXIMUM OF THE γ-MAX-SUBADDITIVE FUNCTIONALS ON A CLASS OF TOPOLOGICAL GROUPS

It is shown that if a 'max-subadditive funtional' p(x) defined on some symmetric neighborhood U0 of zero vector θ in a 'b.f.-toplological group' X is 'upper semi-cotinuous' at a point x0 ∈ U0, or 'lower semi-continuous' in some neighborhood V(x0) U0 and X is of second category; then p(x) can attain its supremum in U0. And there is a similar conclusion for the γ-max-subadditive functional when its supremum is 0 and if U0 is 'pseudo-bounded' set in X.

ALMOST PERIODICITY AND EQUICONTINUITY OF THE TOPOLOGICAL TRANSFORMATION GROUP

In this paper, a characterization of almost periodicity of topological transformation groups on uniform spaces is given. By searching the appropriate base for uniform structure, it is shown that the topological transformation group is topologically equivalent to an isometric one if it is uniformly equicontinuous.

Transversality on locally pseudocompact groups

Two non-discrete Hausdorff group topologiesτandδon a group G are called transversal if the least upper boundτ⋁δofτandδis the discrete topology.In this paper,we discuss the existence of transversal group topologies on locally pseudocompact,locally precompact,or locally compact groups.We prove that each locally pseudocompact,connected topological group satisfies central subgroup paradigm,which gives an affirmative answer to a problem posed by Dikranjan,Tkachenko,and Yaschenko[Topology Appl.,2006,153:3338–3354].For a compact normal subgroup K of a locally compact totally disconnected group G,if G admits a transversal group topology,then G/K admits a transversal group topology,which gives a partial answer again to a problem posed by Dikranjan,Tkachenko,and Yaschenko in 2006.Moreover,we characterize some classes of locally compact groups that admit transversal group topologies.

A Unified Convergence Theorem

We prove a unified convergence theorem, which presents, in four equivalentforms, the famous Antosik-Mikusinski theorems. In particular, we show that Swartz' three uniformconvergence principles are all equivalent to the Antosik-Mikusinski theorems.