On submaximal spaces

Properties of spaces in which every subset is open in its closure are investigated. When scattered, such submaximal spaces are nodec (in the sense of van Douwen) and conversely. Every countably compact Hausdorff nodec space is the free topological sum of finitely many Alexandroff compactifications of discrete spaces. Pseudocompact Tychonoff submaximal spaces are shown to be scattered. The authors do not know if there is an infinite regular connected submaximal space, or if there is in ZFC a Hausdorff submaximal topological group, but we prove that every separable Tychonoff submaximal space is totally disconnected. We pay special attention to Lindelِf submaximal spaces, showing, in particular, that pseudo-Lindelِf submaximal spaces are Lindelِf. The last section is devoted to submaximal topological groups, where it is established that each pseudocompact submaximal topological group is finite and each totally bounded submaximal topological group is countable.