Hedgehog frames and a cardinal extension of normality

The hedgehog metric topology is presented here in a pointfree form, by specifying its generators and relations. This allows us to deal with the pointfree version of continuous (metric) hedgehog-valued functions that arises from it. We prove that the countable coproduct of the metric hedgehog frame with k spines is universal in the class of metric frames of weight k.N-0. We then study k-collectionwise normality, a cardinal extension of normality, in frames. We prove that this is the necessary and sufficient condition under which Urysohn separation and Tietze extension-type results hold for continuous hedgehog-valued functions. We show furthermore that k-collectionwise normality is hereditary with respect to F-sigma-sublocales and invariant under closed maps. (C) 2018 Elsevier B.V. All rights reserved.