Notes on remainders of topological spaces in some compactifications

In this paper, we investigate the compactifications of some topological spaces such that their remainders have countable tightness. We also study addition theorems for compacta. The main results are: (1) If bX is a compactification of a first-countable space X   with a GδGδ-diagonal (or a space X   with a point-countable base) and bX∖XbX∖X has countable tightness, then both bX   and bX∖XbX∖X have countable fan-tightness; (2) If a non-locally compact paratopological group G has a compactification bG   such that the remainder bG∖GbG∖G is the union of a finite family of metrizable subspaces, then G   is locally separable and locally metrizable; (3) If a compact Hausdorff space Z=X∪YZ=X∪Y, where X is a non-locally compact topological group which is a σ-space and dense in Z, and Y is a semitopological group, then Z   is separable and metrizable; (4) If a compact Hausdorff space Z=X∪YZ=X∪Y, where X is a non-locally compact paratopological group that has a countable network and is dense in Z, and Y is a semitopological group, then Z is separable and metrizable. Among them (2) and (3) improve the corresponding results given by A.V. Arhangel'skii in [7].