Normality of powers implies compactness

Abstract

In this note we give a short proof for the theorem of N. Noble which asserts that each power of a T1-space being normal implies that the space is compact. Recently N. Noble proved the following result: 1. (NOBLE [N].) If each power of a T,-space is normal, then the space is compact. This remarkable theorem has been applied by Herrlich and Strecker [H-S] and by Franklin, Lutzer and Thomas [F-L-T] to give two different categorical characterizations of the category of compact Hausdorff spaces as a subcategory of the Hausdorff spaces. Noble's original proof derived this theorem as a corollary of a more complicated one which in turn depended on a long chain of previous results of Noble and others. Keesling has given a short and very elegant derivation of Noble's theorem from two well-known theorems, one of Stone (see ?2 below) and one of Morita [K]. Our purpose in this note is to present another short' proof of Noble's theorem which may point the way to an elementary proof, i.e., one which proceeds directly from the definitions without relying on any powerful theorem along the way. Our nonelementary proof relies on theorems of Stone and Glicksberg: 2. (STONE [S].) If a product of T1-spaces is normal, all but at most a countable number of the factor spaces must be countably compact. 3. (GLICKSBERG [G].) If X is completely regular, then the identity map on any pseudocompact power X' extends to a homeomorphism from f3(X') to (/3X) . PROOF OF 1. Let Xm be an uncountable power of X. Since (Xm)'=XI, XI must be countably compact by 2 and is therefore pseudocompact. We Received by the editors December 28, 1971. AMS 1970 subject classqilcations. Primary 54B10, 54D30; Secondary 54D15, 54D35.