Hedgehog frames and a cardinal extension of normality

• Autor(a) principal: Gutiérrez García, Javier
• Data de Publicação: 2019
• Outros Autores: Mozo Carollo, Imanol, Picado, Jorge, Walters-Wayland, Joanne
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
• Texto Completo: http://hdl.handle.net/10316/90471; https://doi.org/10.1016/j.jpaa.2018.08.001
• topic: Frame, locale, frame of reals, metric hedgehog frame, metrizable frame, weight of a frame, separating family of localic maps, universal frame, join cozero \kappa-family, normal frame, \kappa-collectionwise normal frame, closed map

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 \kappa⋅\aleph_0. We then study \kappa-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 \kappa-collectionwise normality is hereditary with respect to F_\sigma-sublocales and invariant under closed maps.