Covering Dimension and Universality Property on Frames

Studies such as Dube et al. (Topology Appl. 160(18):2454–2464 2013), Georgiou et al. (Algebra Universalis 80(2):16, 2019), Iliadis (2005; Topology Appl. 179, 99–110, 2015; Topology Appl. 201, 92–109, 2016) focus on the notion of saturated class of spaces and frames and the existence of universal elements in such classes. Recently, the notion of dimension for frames has been combined with this universality property (Georgiou et al., Algebra Universalis 80(2):16, 2019; Iliadis, Topology Appl. 201:92–109, 2016). In this paper we study such a property, combining it with the notion of the covering dimension dim for frames (Charalambous, J. London Math. Soc. 8(2):149–160, 1974). We define the base dimension like-function of the type dim for frames and, based on the notion of the saturated class of bases, which is inserted and studied in Georgiou et al. (Order 37:427–444, 2020), we prove that in a class of bases which is characterized by this dimension there exist universal elements. Also, we study the notion of saturated class of frames, which is inserted in Georgiou et al. (Algebra Universalis 80(2):16, 2019), proving that in classes of frames which are determined by the covering dimension dim there exist universal elements.