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A short proof is given that metric spaces which have no closed discrete subspaces of measurable cardinal are realcompact. This result is used to obtain the Shirota theorem that topologically complete spaces which have no closed discrete subspaces of measurable cardinal are realcompact. The result stated in the title is usually obtained as a corollary to the Katetov theorem [3] that paracompact spaces with certain cardinal restrictions are realcompact, or the Shirota theorem [5] mentioned in the abstract. Both my proof and Katetov's proof use techniques similar to those used by Marczewski-Sikorski [4], Theorem 6, which implicitly contains a proof of the following theorem due to Stone [6]. THEOREM 1. Every open cover of a metric space has a a-discrete open refinement. For completeness we sketch the proof. LetQ s={A,} ses be an open cover of the metric space (X, d). Let A, ={x: d(x, X-A )>l/2 } for seS and neN. These are clearly open sets. Now suppose < is a well-ordering of S and define Btn=AXt,n-cl(Us<t As, n+l) Then Mn = {Bt, ts is a discrete family of open sets and = UnEN 4n is a a-discrete open refinement of /. For the terminology used below we refer the reader to Chapters 8 and 12 of [2]. THEOREM 2. Let X be a metric space it'hich has no closed discrete subspace of measurable cardinal. Then X is realcompact. If X is not realcompact, there exists a free ultrafilter of zero sets Y with the countable intersection property. Let sd={Int Z(f) :Z(f)oY}. We easily see that Q/ is an open cover of X, so let X= U 4n be a a-discrete open refinement ofQS. Setting An=U {E:Ecin%}, we have n (X-An) = 0, which shows X-AmEIEY for some meN since Y has the countable intersection property. Thus X-Am 'JmFY implies AmFY. Define Ju:g(m)-~{O 1} by yu(Y?)=1 if cl(U YJ)ec; yu(Y?)=O if cl(U Y)09. Received by the editors February 1, 1971. AMS 1970 subject classiflcations. Primary 54D60, 54-02; Secondary 54E15.

A short proof that metric spaces are realcompact