| In this paper, Base-Countably weakly θ refinable spaces and Nearly-Meso compact spaces are introduced, and the closed hereditary, Tychonoff product and mapping properties of these two kinds of spaces are studied. The following main results were obtained:(i) {Fi}i∈N=∪n∈N An is a closed cover of X , for each x∈ X, there existsn∈N , such that 1≤ord(x,An)(?)ω,and each Fi(i∈N) is a closed base-countably weakly θ refinable space relative to X , then X is a base-countably weakly θ refinable space .(ii) Let Y be a base -countably weakly θ refinable space and f:X→Y be a base-countably weakly θ refinable mapping and ω( X) ≥ ω( Y) , if Y is regular base-countably weakly θ refinable space then X is base-countably weakly θrefinable .(iii)Every closed subspace of Nearly mesocompact spaces is nearly mesocompact.(iv)A space X is nearly mesocompact if and only if every monotone open cover U has an open refinement that is compact-finite on some dense subset of X .(V )Let X = Πα∈Λ Xα be a |Λ| - paracompact space , then X is nearly mesocompactspace if and only if Πα∈Λ Xα is nearly mesocompact for each F∈[Λ](?)ω.(vi)A countable compact, nearly mesocompact T3-space is compact. |
