Abstract: The notion of base-submesocompact spaces is introduced and the nature of the question is discussed: if X is a base-submesocompact space, Y is a closed subset of X, and ω(X)=ω(Y), then Y is a base-submesocompact space. Let X be base-submesocompact space and X′ be an F_σ subset of X, then X′ is base-submesocompact reative to X. Under perfect mapping, the inverse image of base-submesocompact space remains base-submesocompact space.

Key words: topology, base, base-submesocompact, perfect mapping