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In this paper, we prove a dichotomy theorem for remainders in compactifications of paratopological groups: every remainder of a paratopological group G is either Lindelöf and meager or Baire. Moreover, we give a negative answer for a question posed by D. Basile and A. Bella in [6], and some questions about remainders of paratopological groups are posed in the paper.

We investigate the relations between decreasing sequences of sets and the insertion of semi-continuous functions, and give some characterizations of countably metacompact spaces, countably paracompact spaces, monotonically countably paracompact spaces (MCP), monotonically countably metacompact spaces (MCM), perfectly normal spaces and stratiable spaces.