二阶算术子系统中的Lebesgue数与Atsuji空间
Lebesgue numbers and Atsuji spaces in subsystems of second order arithmetic
摘要：
在二阶算术子系统的框架下研究完备可分度量空间的性质。特别是我们认为勒贝格和Atsuji空间。前者是使得每个开覆盖U都有一个勒贝格数，即一个正数q，使得对于空间的每个点x，都存在一个U的元素，它包含中心为x、半径为q的球;后者是使得每个连续函数到另一个完全可分度量空间是一致连续的。主要结果是：“每个紧空间是Lebesgue空间”等价于WKL_0，“每个完备Lebesgue空间是紧空间”和“每个完备Atsuji空间是紧空间”等价于ACA_0，“每个Lebesgue空间是Atsuji空间”在RCA_0中可证;“每个Atsuji空间都是Lebesgue”这一命题在ACA_0中是可证明的，但我们不知道它是否等价于ACA_0。证明了“到闭集的距离是一个连续函数“等价于Pi^1_1-CA_0;证明了“存在一个完备的可分度量空间，它是完备的，Heine-Borel紧的”等价于Pi^1_1-CA_0。Lebesgue，Atsuji）”都等价于WKL_0。
英文摘要：
We study properties of complete separable metric spaces within the framework of subsystems of second order arithmetic. In particular we consider Lebesgue and Atsuji spaces. The former are those such that every open covering U has a Lebesgue number, i.e. a positive number q such that for every point x of the space, there exists an element of U which contains the ball of center x and radius q; the latter are those such that every continuous function into another complete separable metric space is uniformly continuous. The main results we obtain are the following: the statement "every compact space is Lebesgue" is equivalent to WKL_0; the statements "every perfect Lebesgue space is compact" and "every perfect Atsuji space is compact" are equivalent to ACA_0; the statement "every Lebesgue space is Atsuji" is provable in RCA_0; the statement "every Atsuji space is Lebesgue" is provable in ACA_0, but we do not know if it is equivalent to ACA_0. We also prove that the statement "the distance from a closed set is a continuous function" is equivalent to Pi^1_1-CA_0; the statements "there exists a complete separable metric space which is perfect and Heine-Borel compact (resp. Lebesgue, Atsuji)" are all equivalent to WKL_0.
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