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这帖举的反例是离散空间和无限维空间, 在ProofWiki有证明: 有限维欧氏空间中 totally bounded等价于bounded. 又见Math.StackExchange. 这个证明只用到了欧氏空间的“open balls are totally bounded”这个性质. 因此可以推广为“If every open ball in $X$ is totally bounded, then every bounded subset of $X$ is totally bounded”.
在Encyclopedia of Math也有这句话: A subspace of a Euclidean space is totally bounded if and only if it is bounded. The converse is not true: An infinite space in which the distance between any two points is one, as well as a sphere and a ball of a Hilbert space, are bounded, but not totally bounded, metric spaces.
在Wikipedia出现了两处: 第一处是 Each totally bounded space is bounded (as the union of finitely many bounded sets is bounded). The reverse is true for subsets of Euclidean space (with the subspace topology), but not in general. For example, an infinite set equipped with the discrete metric is bounded but not totally bounded:[3] every discrete ball of radius $ε = 1 / 2$ or less is a singleton, and no finite union of singletons can cover an infinite set.
第二处是 A subset of the real line, or more generally of finite-dimensional Euclidean space, is totally bounded if and only if it is bounded.[4][3]
这里引用的资料是:
[3] Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6. p.182
[4] Kolmogorov, A. N.; Fomin, S. V. (1957) [1954]. Elements of the theory of functions and functional analysis,. Vol. 1. Translated by Boron, Leo F. Rochester, N.Y.: Graylock Press. pp. 51–3. |
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