In-between theorems in uniform spaces

Authors:
D. Preiss and J. Vilímovský

Journal:
Trans. Amer. Math. Soc. **261** (1980), 483-501

MSC:
Primary 54C30; Secondary 54E15

DOI:
https://doi.org/10.1090/S0002-9947-1980-0580899-0

MathSciNet review:
580899

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Abstract: Necessary and sufficient conditions for the existence of a uniformly continuous function in-between given functions 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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DOI:
https://doi.org/10.1090/S0002-9947-1980-0580899-0

Keywords:
Uniform space,
mixed preimage,
far sets with respect to a cover,
perfectly refinable cover,
Ext-uniformity,
inversion-closed space

Article copyright:
© Copyright 1980
American Mathematical Society