Products on Maximal Compact Frames

Abstract

In topological spaces, many topological properties such as separation properties, paracompactness etc. are preserved under the act of taking product of topological spaces. The well known Tychonoff theorem in topological spaces which states that product of compact spaces is compact. Many of these results can be extended to the “generalized topological spaces”, known as locales (frames). According to Tychonoff product theorem for locales, locale product(coproduct of frames) of compact locales (frames) is compact. In this paper, we examine whether the coproduct of maximal compact frames is maximal compact. We examine the case for a finite coproduct and for an arbitrary coproduct of maximal compact frames. Every subframe of a compact frame is compact but a sublocale need not be. We provide a characterization for a sublocale of a maximal compact frame to be a maximal compact sublocale.