On submaximal spaces
``Locally closed sets, sets open in their closure, play an enigmatic role in topology. Their appearance is often surprising and a general theory is elusive: one only needs to note that locally compact subsets of Hausdorff spaces and connected subsets of the real line have the property.'' With this introduction, the authors begin a detailed and extremely well written investigation of this property. Included is a lengthy list of references which, in combination with this article, will get a novice in the study of this property off to a great start. Some of the results are: (1) when scattered, such submaximal spaces are nodec (in the sense of van Douwen) and conversely; (2) Every countably compact Hausdorff nodec space is the free topological sum of finitely many Alexandroff compactifications of discrete spaces; (3) Pseudocompact Tychonoff submaximal spaces are scattered; (4) Pseudo-Lindelöf submaximal spaces are Lindelöf. The last section is the study of submaximal topological groups where it is shown that a pseudo-compact submaximal topological group is finite and a totally bounded submaximal topological group is countable. The authors admit that their study does not reveal all as unanswered questions start appearing early in section 1 with the first being ``Is every submaximal space \(\sigma\)-discrete?'' Another is ``Is there an infinite regular connected submaximal space?'' The reviewer wishes that every paper was written as well as this one. The writing makes it both easy and enjoyable to read. I hope that any of you read and enjoy as I did.
pseudo-compact submaximal topological group
totally bounded submaximal topological group
scientific article; zbMATH DE number 792536
scientific article