拓扑空间和度量空间中函数的逼近系统
Approximation systems for functions in topological and in metric spaces
摘要：
可计算性的TTE方法的一个显着特征是通过无穷名称来表示参数值和相应的函数值。文件指出了在某些情况下消除使用这类名称的两种方法。第一个是拓扑空间的情况下，选定的索引可数基地。假设从一个这样的空间到另一个这样的空间的部分函数，其所选基具有递归可重复的指标集，并假设第一个空间中的基开集的交集在Weihrauch-Grubba意义下是可计算的。然后，通过包含自变量值的基集的指标与包含相应函数值的基集的指标之间存在适当的递归可积关系，刻画了函数的普通TTE可计算性，这个结果可以看作是Korovina和Kudinov的一个结果的改进.第二种方法适用于具有选定索引可数稠密子集的度量空间。如果从一个这样的空间到另一个这样的空间给出一个部分函数，那么，在关于这些空间的半可计算性假设下，该函数的普通TTE可计算性的特征在于存在一个适当的递归四元组集。它们中的任何一个都包括来自第一空间中的所选稠密子集的元素的索引、对该元素与自变量值之间的距离的有理界限进行编码的自然数、来自第二空间中的所选稠密子集的元素的索引以及对该元素与函数值之间的距离的有理界限进行编码的自然数。文中的一个例子表明，利用无穷名消去法第一种方法，可以简单地刻划实函数的可计算性。
英文摘要：
A notable feature of the TTE approach to computability is the representation of the argument values and the corresponding function values by means of infinitistic names. Two ways to eliminate the using of such names in certain cases are indicated in the paper. The first one is intended for the case of topological spaces with selected indexed denumerable bases. Suppose a partial function is given from one such space into another one whose selected base has a recursively enumerable index set, and suppose that the intersection of base open sets in the first space is computable in the sense of Weihrauch-Grubba. Then the ordinary TTE computability of the function is characterized by the existence of an appropriate recursively enumerable relation between indices of base sets containing the argument value and indices of base sets containing the corresponding function value.This result can be regarded as an improvement of a result of Korovina and Kudinov. The second way is applicable to metric spaces with selected indexed denumerable dense subsets. If a partial function is given from one such space into another one, then, under a semi-computability assumption concerning these spaces, the ordinary TTE computability of the function is characterized by the existence of an appropriate recursively enumerable set of quadruples. Any of them consists of an index of element from the selected dense subset in the first space, a natural number encoding a rational bound for the distance between this element and the argument value, an index of element from the selected dense subset in the second space and a natural number encoding a rational bound for the distance between this element and the function value. One of the examples in the paper indicates that the computability of real functions can be characterized in a simple way by using the first way of elimination of the infinitistic names.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
