Concepts of general topology in constructive mathematics and in sheaves, II
Concepts of general topology in constructive mathematics and in sheaves, II
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构造数学和滑轮中的一般拓扑概念，II
DOI：
10.1016/0003-4843(82)90010-9
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发表时间：
1981
期刊：
影响因子：
--
通讯作者：
R. Grayson
中科院分区：
文献类型：
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作者：
R. Grayson
本文在某种程度上是Grayson [6]的续篇，以下简称CGT I。人们应该与CGT我，或至少有机会获得它，因为经常提到它。这里的重点是更多的（层）模型和较少的建设性理论的拓扑结构;与CGT I的另一个不同之处在于，在这里用于对模型进行推理的方法通常是（必然）经典的，而在CGT I中，我们关心的是尽可能建设性地对它们进行推理。在第2节中，我们提出了一类更一般的模型，即丛的截面空间;在CGTI中，我们只考虑了这些“常数”的例子。表示定理2.1. [5]的第1部分表明，所有内部拓扑空间在“拓扑模型”（即，在拓扑层空间）出现的子空间，这种空间的部分，因此，这类模型有一个内在的利益，和大部分的文件是有关建立这些空间的内部性质之间的联系和外部性质的纤维。我们的主要工具是扩展引理（第2.3节），它的第一个应用包括一个内部表征的子集的空间的部分是'代表'的外部子集的th~ bundle（第2.4节）。在第3节和第4节中，我们考虑紧性和连通性的概念，并在映射空间中解释它们。这导致了两个主要的结果，有效性的所有空间在所有的拓扑模型，两个定理的连通和链连通度量空间。我们承认这里存在差距，因为我们没有
This paper is to some extent a sequel to Grayson [6], hereafter referred to as CGT I. One should be tamiliar with CGT I, or at least have access to it, since frequent reference is made to it. The emphasis here however is more on the (sheaf) models and less On the constructive theory of topology; another difference from CGT I is that the methods used here in reasoning about the models are often (necessarily) classical, whereas in CGT I we were concerned to reason about them constructively as far as possible.In the present section we cover some preliminary topological notions. In Section 2 a more general class of models is presented, the spaces of sections of bundles; in CGTI we only considered'constant'examples of these. The Representation Theorem 2.1. 1 of [5] shows that all internal topological spaces in'topological models'(ie, in topoi c~ sheaves over spaces) arise as subspaces of such spaces of sections; thus this clat,~ of models has an intrinsic interest, and the bulk of the paper is concerned with establishing connections between internal properties of these spaces and external properties of their fibres. Our main tool is the Spreading Lemma (Section 2.3), whose first applications include an internal characterisation of subsets of a space of sections which are'representable'by external subsets of th~ bundle (Section 2.4). In Sections 3 and'4 we consider notions of compactness and connectedness, and interpret them in spaces of~ ections. This leads to two main results, on the validity for all spaces in all topological models, of two theorems on connected and chain-connected metric spaces. We acknowledge a gap here, in that we have not