1. Introduction. Prof. K. Morita [4 has introduced the notion of M-spaces. He calls a topological space X an M-space if there exists a normal sequence {1I i-1, 2, ...} of open coverings of X satisfying the condition (.) below: (If a family consisting of a countable number of subsets Jof X has the finite intersection property and contains as a (*) member a subset of S(x0, 1I) for every i and for some fixed point x0 of X, then {/ K } =/= . Recently, T. Kand5 2J has proved the following theorem.

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... eorem 1. Let {A.} be, a locally finite covering of a Hausdorff space X and let each A be a closed G-subset of X. If each A is a normal M-space with respect to its relative topology, then the whole space X is also a normal M-space. In this connection he raised a problem whether Theorem i is valid without the G-condition for A [2, p. 1053. The purpose of this note is to give an affirmative answer to this problem; namely, we shall prove the following theorem. Theorem 2. Le {A} be a locally finite closed covering of a Hausdorff space X. If each A is a normal M-space with respect to its relative topology, then the whole space X is also a normal M-space. Most terminologies and notations used in this note are the same as those of J. W. Tukey 7. We are indebted to Prof. K. Morita for valuable advices and encouragements throughout this study. 2. Lemmas. Lemma 1. Let {Ai i-1, 2} be a binary closed covering of a Hausdorff space X. If each Ai is a normal M-space, then X is a normal M-space. Proof. According to a result of A. 0kuyama [6 each A is collectionwise normal and countably paracompact, and hence by K. Morita 5_ the whole space X is also collectionwise normal and countably paracompact. Suppose that (A) (A.)-.) Then we have (A) (A) =X. ) Since X is normal there exist two closed G-subsets F and 1) 3(A) means the boundary of a set A. 2) 3(A) means the interior of a set A.