Generalized monotonically T 2 spaces
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GENERALIZED MONOTONICALLY T2 SPACES W. H. SUN School of Control Science and Engineering, Shandong University, Jinan, Shandong, 250061, China School of Mathematics and Statistics, Shandong University, Weihai, Shandong, 264209, China e-mail: [email protected] (Received June 17, 2019; revised April 24, 2020; accepted May 4, 2020)
Abstract. The notion of μ-monotonically T2 spaces is introduced, which is a generation of monotone T2 separation in general topological spaces. The characterization and some properties of μ-monotonically T2 spaces are given. Besides, the definition of the box product of the topologies is extended to generalized topologies, and we prove that the box product of μ-monotonically T2 spaces is μ-monotonically T2 .
1. Introduction Generalized topologies were introduced by E. H. Moore, [22, pp. 53–80], both in the closure operator and the closed sets forms. For a long time this was considered as a part of algebra. E.g., in P. M. Cohn [3] this definition was repeated. In A. G. Kurosh [15, Ch. VI, §7], this definition was extended from the power set of a set to any partially ordered set, which was repeated in Encyclopedia of Math. [12, Vol. 2, p. 167]. A. Appert [1] considered some generalizations of topologies, among others omitting the property that the intersection of two open sets should be open. The same concept, i.e., generalized topology, was independently invented by F. W. Levi [16] and T. S. Motzkin [23], under the name “onvexity structures”. They had the aim to prove elementary facts of convexity in finite dimensional Euclidean spaces in the broader context of a closure operator on any set, rather than convex hulls, or convex sets, in finite dimensional ∗ This research is supported by Natural Science Foundation of Shandong Province(Grant No. ZR2019MA051) and National Natural Science Foundation of China–Shandong joint fund (No. U1806203) and the Fundamental Research Funds for the Central Universities (No. 2019ZRJC005) and National Natural Science Foundation of China (Grant No. 11501328). Key words and phrases: generalized topological space, μ-monotonically T2 space, box product of generalized topologies. Mathematics Subject Classification: 54A05, 54C08, 54D10.
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Euclidean spaces. These investigations were led further by D. C. Kay and E. W. Womble [14]. Products of “convexity structures” were defined by J. Eckhoff [11], which differ from Cs´asz´ ar’s product, but coincide with the product in the sense of category theory, as later pointed out by E. Makai, Jr., I. Peyghan and B. Samadi [20]. A monograph appeared about the results of “axiomatic convexity theory” till its writing, by V. P. Soltan [26]. The second formal reintroduction of generalized topological spaces to topology is due to S. Lugojan [18] and A. S. Mashhour, A. A. Allam, F. S. Mahmoud and F. H. Khedr [21]. Actually they defined only strong generalized topological spaces, under the names “generalized topological spaces” and “supratopological spaces”. Of t
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