The Dimension of the Rational Points in Hilbert Space
The Dimension of the Rational Points in Hilbert Space
复制标题
希尔伯特空间中有理点的维数
DOI：
10.2307/1968851
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发表时间：
1940
影响因子：
4.9
通讯作者：
P. Erdös
中科院分区：
文献类型：
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作者：
P. Erdös
其中ni是正整数。设R1 = R0。显然是R0 C R1 C r定理！Dim R 0 = Dim R1 = Dim R = 1。在我们继续证明之前让我们注意到笛卡尔积R1 X R1同胚于R1。由此得到存在一个度量可分完全空间X，使得dim X = dim X X X = 1。这似乎是对量纲理论的“乘积问题”的一个新贡献。值得注意的是R1在它的任意两个点之间是不相连的。证明暗淡的R0 > 0。设U是直径小于2的H的开子集，使得UR0。因此，设(2)属于u。我们将定义一个自然数序列m l， M2，使得
where ni are positive integers . Let R1 = R0 . Clearly R0 C R1 C R. THEOREM! Dim R 0 = dim R1 = dim R = 1 . Before we proceed with the proof let us remark that the Cartesian product R1 X R1 is homeomorphic to R1 . Hence we obtain that There exists a metric separable complete space X such that dim X = dim X X X = 1 . This seems to be a new contribution to the "product problem" of the theory of dimensions . It might also be worth noticing that R1 is disconnected between any two of its points . Proof that dim R0 > 0 . Let U be an open subset of H of diameter less than 2 and such that UR0 0 . Let therefore (2) belong to U. We shall define a sequence of natural numbers m l , M2, such that