论文标题
Borel可测量的Hahn-Mazurkiewicz定理
Borel Measurable Hahn-Mazurkiewicz Theorem
论文作者
论文摘要
由于Hahn和Mazurkiewicz众所周知，每个Peano连续体都是单位间隔的连续图像。我们证明，可以以borel可测量的方式实现的连续映射作为输入，并作为输出作为输出的连续映射。同样，我们找到了一个可衡量的bore尺寸分配，该分配需要任何非空的紧凑型公制空间，并将连续的映射从康托尔设置分配到该空间。为此，我们使用伯吉斯选择定理。最后，发现了一种可以测量的弧形方法，该方法可以在Peano Continuum中连接两个选定点的弧线。
It is well known due to Hahn and Mazurkiewicz that every Peano continuum is a continuous image of the unit interval. We prove that an assignment, which takes as an input a Peano continuum and produces as an output a continuous mapping whose range is the Peano continuum, can be realized in a Borel measurable way. Similarly, we find a Borel measurable assignment which takes any nonempty compact metric space and assigns a continuous mapping from the Cantor set onto that space. To this end we use the Burgess selection theorem. Finally, a Borel measurable way of assigning an arc joining two selected points in a Peano continuum is found.
