E-compactness in pointfree topology

Bibliography: leaves 100-107.The main purpose of this thesis is to develop a point-free notion of E-compactness. Our approach follows that of Banascheski and Gilmour in [17]. Any regular frame E has a fine nearness and hence induces a nearness on an E-regular frame L. We show that the frame L is complete with respect this nearness iff L is a closed quotient of a copower of E. This resembles the classical definition, but it is not a conservative definition: There are spaces that may be embedded as closed subspaces of powers of a space E, but their frame of opens are not closed quotients of copowers of the frame of opens of E. A conservative definition of E-compactness is obtained by considering Cauchy completeness with respect to this nearness. Another central notion in the thesis is that of K-Lindelöf frames, a generalisation of Lindelöf frames introduced by J.J. Madden [59]. In the last chapter we investigate the interesting relationship between the completely regular K-Lindelöf frames and the K-compact frames

Local connectedness of frames

In this thesis, we undertake a systematic study of local connectedness of frames. Among other central ideas in this study is that of a connected congruence on a frame. We show that the two definitions of a connected congruence in literature (section 2.2) are not equivalent, and hence introduce a new term for one of them. We also prove that, using Baboolal's methods, if the Stone-Cech compactification βL is locally connected then L need not be locally connected for completely regular frame L. This happens in chapter 5