We introduce and study some new covering properties in frames that are pointfree variants of generalized compactness notions. We establish the concept of star-finite covers on a frame and introduce Hypocompact frames. (Co)refecting such frames via nearness completion [1] is proposed. Variants of point-finite covers ([2, 3, 4]) and associated variants of pointfree are studied. The talk is an exposition of these new concepts and work in progress with Mark Sioen (Vrije Universiteit Brussels, Belgium).
References
[1] B. Banaschewski, Completion in pointfree topology, Lecture Notes in Math. and Appl. Math., Univ. of Cape Town, SoCat 94, No2/1996.
[2] C.H. Dowker and D. Strauss, Paracompact frames and closed maps, Symp. Math. 16
(1975), Convegno sulla Topologia Insiemisticae Generale, 93 - 116.
[3] I. Naidoo, D. Baboolal and R.G. Ori, Metacompact nearness frames, Math. Japonica 52(1)(2000), 45 - 49.
[4] M.J. Ferreira and J. Picado, On Point-finiteness in Pointfree Topology, Appl. Categ. Structures (2007) 15: 185 - 198.
[5] J. Picado and A. Pultr, Frames and Locales: topology without points, Birkh¨auser–Springer, Basel, 2012.