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Abstract.Necessary and sufficient conditions for the existence of a uniformly continuous function in-between given functions / > g on a uniform space are studied. It appears that the investigation of this problem is closely related to some combinatorial properties of covers and leads to the concept of perfect refinability, the latter being used, e.g., to obtain an intrinsic description of uniform real extensors. Several interesting classes of uniform spaces are characterized by special types of in-between theorems. As examples of applications we show that the usual in-between theorems in topology and their generalizations, as well as some important methods of construction of derivatives of real functions, follow easily from the general results.0. Introduction. The long history of extension and in-between theorems led to the nice results of M. Katëtov and E. Michael for topological spaces. It appears that in some applications of continuous structures the topological point of view is too restrictive. This, as well as the natural development of in-between problems, led us to formulate and prove the uniform version of these theorems. Although our approach involves quite different ideas, the topological results become easy corollaries. To some extent our ideas are of the obvious sort, but at least one "unstable" property, perfect refinability, ("unstable" here means with respect to usual uniform operations with covers) comes in, in a rather essential way. The results show that the strange definition of perfect refinability is a quite good approximation to a key idea in this subject. Besides the general theory contained in the first two paragraphs we show some interesting applications to topology and the theory of real functions.Throughout this paper we will refer to [7] for basic definitions and results pertaining to uniform spaces. All uniform spaces are supposed to be Hausdorff. We shall denote by R the real line with its usual metrizable uniformity and usual order, 7? will denote the space of extended reals, i.e. the uniform sum Ä V{_0°}V { + 00} with the usual order -00 < r < +00 for all r G R. In the text we often simply write function instead of /^-valued function. For a positive real number 5 we shall denote by 9(5) the uniform cover {

Abstract. Values of the compact interval and other spaces under coreflectors in the category of uniform spaces are studied. It is shown that any coreflector which changes the usual uniformity of the interval produces a new uniformity which contains all finite Baire partitions of the interval.

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