Abstract: The notion of base countable metacompact space is introduced through researching the relation between base countable metacompact space and its subspace. From the research, it draws the conclusion that basenormal countable metacompact space is base countable metacompact space, and it proves that open closed finite to 1 map preserves base countable metacompactness.

Key words: countable metacompact space；basecountable metacompact space；basecountable metacompact space relative to X；basenormality