ON CAUCHY HOMOMORPHISMS OF NEARNESS FRAMES

To include the pointfree case, the notion of a Cauchy map easily exten ds as the condition that the preimage of any regular Cauchy filter con tains a regular Cauchy filter. This extension, however, can be unsatis factory if the regular Cauchy filters are scarce or non-existent, maki ng the condition too weak, indeed sometimes even void. In this paper a variant of the notion, independent on the existence of any kind of fi lters, is studied: a homomorphism is called fully Cauchy if it lifts t o the completions. This is generally stronger than, and in the spatial and metric cases it coincides with the previously mentioned property. Moreover, it is stable under localization.