Remainders Of Topological Spaces And The Generalized Metrizability Properties

In this thesis, remainders of topological spaces (includes paratopological groups, semitopological groups and homogeneous spaces) in some compactifications are stud-ied. We mainly investigate the relationship between the generalized metrizability properties of remainders and the generalized metrizability properties of the space it-self. In addition, we also discuss K-spaces and the homeomorphisms of topological groups.In Chapter1, some background, main results and preliminaries are given. In Section1, we recall the research progress of remainders of topological spaces at home and aboard, and offer the main results in this thesis. In Section2, we introduce some preliminary knowledge, including some important concepts and related propositions.In Chapter2, we investigate remainders of general topological spaces and paratopological groups, two theorems by A.V. Arhangel’skii are improved. The fol-lowing two main results are established:(a) If bG is a compactification of a non-locally compact paratopological group G, then Y=bG\G is locally a p-space if and only if at least one of the following conditions holds:(1) G is a Lindelof p-space;(2) G is σ-compact.(b) If bX is a compactification of a nowhere locally compact space X with locally a Gδ-diagonal such that Y=bX\X is a paracompact p-space, then X is separable and metrizable.In Chapter3, we study remainders of semitopological groups and rectifiable spaces, several results given by A.V. Arhangel’skii and Chuan Liu are improved and generalized. Main results are as follows:(c) If a non-locally compact semitopological group G has a compatification bG such that Y=bG\G has locally a point-countable base, then both G and bG are separable and metrizable.(d) If a non-locally compact locally paracompact rectifiable space X has a com-pactification bX satisfying that Y=bX\X has locally a Gδ-diagonal, then X and Y are both have countable bases.In Chapter4, K-spaces and CS-spaces are discussed. An open problem posed by V.I. Malykhin and G. Tironi in [62] is answered firmly: (e) Must a compact K-space have countable tightness?We also investigate K-topological groups and the mapping properties of CS-spaces.In Chapter5, we investigate the homeomorphisms of topological groups, and two open problems posed by A.V. Arhangel’skii and M. Tkachenko in [33] are solved:(f) Is every connected topological group homeomorphic to an w-narrow topolog-ical group?(g) For a zero-dimensional topological group G with neutral element e, must X=G\{e} be homeomorphic to a topological group?...