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On submaximal spaces. (English) Zbl 0826.54002
“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.

MSC:
54A10 Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)
54D30 Compactness
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