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R-SUPERCONTINUOUS FUNCTIONS

机译:R超连续功能

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摘要

A new strong variant of continuity called 'R-supercontinuity' is intro-duced. Basic properties of R-supercontinuous functions are studied and their place in the hierarchy of strong variants of continuity that already exist in the literature is elaborated. It is shown that R-supercontinuity is preserved under the restriction, shrinking and expan-sion of range, composition of functions, products and the passage to graph function. The class of R-supercontinuous functions properly contains each of the classes of (i) strongly 0-continuous functions introduced by Noiri and also studied by Long and Herrington; (ii) D-supercontinuous functions; and (iii) F-supercontinuous functions; and so include all z-supercontinuous functions and hence all clopen maps (≡ cl-supercontinuous func-tions) introduced by Reilly and Vamnamurthy, perfectly continuous functions defined by Noiri and strongly continuous functions due to Levine. Moreover, the notion of r-quotient topology is introduced and its interrelations with the usual quotient topology and other variants of quotient topology in the literature are discussed. Retopologization of the do-main of a function satisfying a strong variant of continuity is considered and interrelations among various coarser topologies so obtained are observed.
机译:引入了一种新的强连续性变体,称为“ R-超连续性”。研究了R超连续函数的基本性质,并阐述了它们在文献中已经存在的连续性强变体的层次结构中的位置。结果表明,R超连续性在范围,函数组成,乘积和图形函数传递的约束,缩小和扩展下得以保留。 R-超连续函数的类适当地包含(i)由Noiri引入并由Long和Herrington研究的强0-连续函数的每个类; (ii)D超连续函数; (iii)F-超连续函数;因此包括所有z超连续函数,因此包括Reilly和Vamnamurthy引入的所有clopen映射(≡cl超连续函数),Noiri定义的完全连续函数以及Levine提出的强连续函数。此外,介绍了r商拓扑的概念,并讨论了它与常用商拓扑和文献中商拓扑的其他变体的相互关系。考虑了满足连续性强烈变化的函数域的重新拓扑,并观察到了如此获得的各种较粗糙拓扑之间的相互关系。

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