Topology with monoidal control

Abstract

This paper defines monoidal control, which is a specialization of control by entourage as presented in [5]. It is shown that on metric spaces monoidal control generalizes bounded control, and describes continuous control. A systematic way of obtaining results from bounded control, in the sense of [1], as results of monoidal-, and hence continouos-, control, is developed. Especially this provides versions of the Hurewicz and Whitehead theorems with monoidal control, thus simultaneously establishing them for continouos control over metric spaces.