Tikhonov semifields and certain problems in general topology

In this article we give a detailed presentation of a number of results connected with the problem of the measurability of cardinals and their topological equivalents (stated in the language of Čech compactifications and Hewitt spaces). The presentation is directed from a single view point by a systematic use of Tikhonov semifields (that is, the Tikhonov products of a certain number of copies of the real line). At the end we examine the “geometrical” properties of Tikhonov semifields based on Gleason's results on the representation of functions on an uncountable direct product in the form of a composition. In particular, we give the result of Gel'fand and Fuks that on the “sphere” of a Tikhonov semifield any continuous real function is constant.