@UNPUBLISHED{publication1, title = "Hedgehog frames and a cardinal extension of normality", author = "{GUTI{\'{E}}RREZ GARC{\'{I}}A, Javier} and {MOZO CAROLLO, Imanol} and {PICADO, Jorge} and {WALTERS-WAYLAND, Joanne}", year = "2018-01-31", abstract = " 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 \κ spines is universal in the class of metric frames of weight \κ\·\ℵ0. We then study \κ-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 \κ-collectionwise normality is hereditary with respect to F-sublocales and invariant under closed maps. ", url = "http://www.mat.uc.pt/preprints/eng\_2018.html", note = "" }