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In [1], Tamaño asked whether or not a space which is the union of a closure-preserving family of compact subsets must be paracompact. It is the purpose of this paper to present a space which admits such a cover, is completely regular and T2, has a basis consisting of open and closed sets, yet fails to be even normal. In this paper, we present an example of a completely regular, 72 space which has a closure-preserving cover consisting of compact sets, yet fails to be normal. Let W={o\cj<.m1} be the collection of countable ordinals. Let 7={(ff, X)\o, XeW, o<X}. Let X= WkjT. We define a topology on A" as follows: For each oeW, a set Vcontaining a shall be a basic open set about a provided : (1) F contains no ordinal other than a itself. (2) V contains all but finitely many members of the set {(o-, X) X>o}. (3) V contains all but finitely many members of the set {(X, a) /<cr}. For each (er, X)eT, the singleton set {(a, X)} is open. It is clear that the sets described above constitute a basis for a topology on X. This topology is completely regular and 72 and, in fact, has a basis consisting of open and closed sets. To see this, let F be a basic open set about an ordinal at. The only candidates for limit points of V, not already in V, are other ordinal numbers. Let a2eW, with a^ax, and suppose af>ax. Let E/={cr2}U {(X, o2)\X<o2, X7^of}KJ{(a2, X)\X>c2}. Then U is a basic open set about o2, and UC\V=0. For if UnV¿¿0, this would require either an ordinal X>o2 such that (a2, X)eVnU, which is not possible, since every member of V has first co-ordinate o-l5 or UC\Vj¿0 would require an ordinal X<a2 such that (X, o2)eVr\U. Since every member of V has first co-ordinate 0*1, this would mean that (X, a2)=(o1, cf2), whence, X=o1. But (o1, o"2) is not a member of U, so that LTiV=0. Thus no ordinal o2~>ox, can be a limit point of V. For o'2<o"i, let t/={o-2}U {(A, o-2)|a<<j2}u{((72, X)\X>ö2, X-^úf\. The same type of argument as Received by the editors March 29, 1971. AMS 1970 subject classifications. Primary 54G20; Secondary 54D20.

A nonparacompact space which admits a closure-preserving cover of compact sets