Abstract: In this paper the notation of base-mesocompactness is introduced and the following results are mainly obtained: (1) Let X be base-mesocompact and X’ an F σ subset of X. If X is normal, then X’ is base-mesocompact relative to X. (2) Let f: X → Y be a base-mesocompact mapping, ω(X) be a regular cardinality of X and ω(X) ≥ω(Y). If Y is base-mesocompact and regular, then X is base-mesocompact. (3) Let f: X → Y be a closed lindel of mapping with regular domain and regular range. If Y is base-mesocompact, then X is base-mesocompact. (4) Let X be base-mesocompact. If Y is locally compact and base-esocompact, then X × Y is base-mesocompact. 

Key words: base-mesocompact, base-paracompact mapping, base-mesocompact mapping;
closed lindelof mapping