In-between theorems in uniform spaces

Preiss, D.; Vilímovský, J.

doi:10.1090/S0002-9947-1980-0580899-0

In-between theorems in uniform spaces
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by D. Preiss and J. Vilímovský PDF

Trans. Amer. Math. Soc. 261 (1980), 483-501 Request permission

Abstract:

Necessary and sufficient conditions for the existence of a uniformly continuous function in-between given functions $f \geqslant g$ 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.

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