A. FWARU, T. DUBE AND M. MUNYWOKI

Abstract

Let L be a completely regular locale. L is called countably com- pactifiable (abbreviated γ-compactifiable) if there exists a countably compact completely regular locale M such that Lis a dense sublocale of M, and every countably compact closed sublocale of Lis closed in M. This class of locales properly encompasses the class of locales of open sets of Tychonoff spaces with analogous features introduced by K.Morita (1973) in Countably-compactifiable spaces. Morita established that if well-separated spaces Xand Ywith count- ably compactifications Rand S, respectively, and the product space R× S is countably compactifiable, then the product space X× Yinherits this property. Our aim is to extend Morita’s findings to locales and frames.

Let βLdenote the Stone-C^ˇech compactification of L. We explore various

operations that preserve the countably compactifiable property, we show that If a locale L is γ-compactifiable, then there is a γ-compactification S of L such that L ⊆ S ⊆ βL . Additionally, we address a fundamental question: Can the product of two countably compact (or countably compactifiable) locales be γ-compactifiable ?