ON SOME METRIZABLE TOPOLOGIES FOR WEAKLY ALMOST PERIODIC FUNCTIONS

Abstract

The article deals with studying metrizable topologies on the additive group of real numbers that compactify this group. The introduced metrizable topology is weaker than the original natural topology on the real axis. It is a modification of the Marchenko topology. In the introduced topology a countable system of neighborhoods is selected based on the spectrum of a given function. An invariant metric is constructed which defines an equivalent topology. The completed metric space is proved to be compact. A pseudometric using only the spectrum of a given scalar almost periodic function is considered. To obtain the Hausdorff space we pass to a factor space. On the factor space the pseudometric is a metric and it is shown that the values of a scalar almost periodic function on the original space coincide with those on the factor space. It is also proved that the set of scalar almost periodic functions on the axis coincides with the set of functions defined on a metric space, which are scalar uniformly continuous in this topology.