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Compactness and Continuity, Constructively Revisited

机译：紧凑性和连续性，建设性地重新审视

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In this paper, the relationships between various classical compactness properties, including the constructively acceptable one of total boundedness and completeness, are examined using intuitionistic logic. For instance, although every metric space clearly is totally bounded whenever it possesses the Heine-Borel property that every open cover admits of a finite subcover, we show that one cannot expect a constructive proof that any such space is also complete. Even the Bolzano-Weierstraβ principle, that every sequence in a compact metric space has a convergent subsequence, is brought under our scrutiny; although that principle is essentially nonconstructive, we produce a reasonable, classically equivalent modification of it that is constructively valid. To this end, we require each sequence under consideration to satisfy uniformly a classically trivial approximate pigeonhole principle―that if infinitely many elements of the sequence are close to a finite set of points, then infinitely many of those elements are close to one of these points―whose constructive failure for arbitrary sequences is then detected as the obstacle to any constructive relevance of the traditional Bolzano-Weierstrafi principle.
机译：在本文中，使用直觉逻辑检查了各种经典紧实度属性之间的关系，包括总有界性和完整性中建设性可接受的一个。例如，尽管每一个度量空间只要具有每个开放封面都承认的有限子封面所具有的Heine-Borel属性，就明确地是完全有界的，但我们表明，人们不能期望得到一个建设性的证据证明任何此类空间也是完整的。甚至在紧凑的度量空间中的每个序列都具有收敛子序列的Bolzano-Weierstraβ原理，也受到了我们的审查。尽管该原则本质上是非建设性的，但我们对其进行合理，经典等效的修改是具有建设性意义的。为此，我们要求所考虑的每个序列都一致地满足经典的琐碎近似鸽孔原理-如果该序列中的无限多的元素接近一组有限的点，那么这些元素中的无限多个都接近这些点中的一个``然后将其对任意序列的建设性失败视为对传统Bolzano-Weierstrafi原理的任何建设性相关性的障碍。

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