可伸缩拓扑中的模糊连续性
Fuzzy Continuity in Scalable Topology
摘要：
在这项工作中，拓扑空间丰富了额外的结构，以提供一个更现实的现实生活中的现象和计算过程的表示，并在同一时间，提供利用强大的技术开发的拓扑结构。所建议的方法是基于拓扑空间X中的不连续结构或尺度Q的概念。此结构用作X上初始拓扑的标度。科学和工程问题需要这样的尺度，因为所有的测量和大多数计算都是在一个确定的尺度上进行的，可扩展性问题对于物理理论和计算算法都很重要。设一个具有不连续结构Q的拓扑空间X到具有不连续结构R的拓扑空间Y的映射，定义了该映射的（Q，R）-连续、弱（Q，R）-连续和R-连续.新古典分析中研究的模糊连续函数是R-连续映射的例子。得到了标度、Q-开集、Q-闭集和（Q，R）-连续映射的不同性质。
英文摘要：
In this work, topological spaces are enriched by additional structures in order to give a more realistic representation of real life phenomena and computational processes and at the same time, to provide for utilization of the powerful technique developed in topology. The suggested approach is based on the concept of a discontinuity structure or scale Q in a topological space X. This structure serves as a scale for the initial topology on X. Problems of science and engineering need such a scale because all measurements and the majority of computations are performed in a definite scale and problems of scalability are important both for physical theories and computational algorithms. Taking a mapping of a topological space X with the discontinuity structure Q into a topological space Y with a discontinuity structure R, we define (Q,R)-continuity, weak (Q,R)-continuity, and R-continuity of this mapping. Fuzzy continuous functions, which are studied in neoclassical analysis, are examples of R-continuous mappings. Different properties of scales, Q-open, Q-closed sets, and (Q,R)-continuous mappings are obtained.
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