Characterizations of intervals via continuous selections
Characterizations of intervals via continuous selections
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通过连续选择来表征区间
DOI：
10.1007/bf02977032
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发表时间：
1997
影响因子：
1
通讯作者：
D. Shakhmatov
中科院分区：
文献类型：
--
作者：
T. Nogura;D. Shakhmatov
我们证明：（i）具有连续选择的路径连通的豪斯多夫空间同胚于以下四个空间之一：单例、[0,1）、[0,1]或长线L，（ii）具有连续选择的局部连通（豪斯多夫）空间必须是可有序的，以及（iii）无限连通的豪斯多夫空间当且仅当它是紧且可有序的时，才具有恰好两个连续选择。我们使用这些结果通过连续选择给出区间的各种特征。例如，(iv) 拓扑空间 X 同构于 [0,1]，当且仅当 X 是无限、可分、连通、豪斯多夫空间，并且恰好有两个连续选择，并且 (v) 拓扑空间 X 同胚于 [0,1)，当且仅当下列等价条件之一成立： (b)X 是无限的、可分的、局部连通的并且恰好有一个连续选择； (c)X 是无限的、度量的、局部连通的并且只有一个连续选择。展示的三个例子证明了我们的结果中各种假设的必要性。
We prove that: (i) a pathwise connected, Hausdorff space which has a continuous selection is homeomorphic to one of the following four spaces: singleton, [0,1), [0,1] or the long lineL, (ii) a locally connected (Hausdorff) space which has a continuous selection must be orderable, and (iii) an infinite connected, Hausdorff space has exactly two continuous selections if and only if it is compact and orderable. We use these results to give various characterizations of intervals via continuous selections. For instance, (iv) a topological spaceX is homeomorphic to [0,1] if (and only if)X is infinite, separable, connected, Hausdorff space and has exactly two continuous selections, and (v) a topological spaceX is homeomorphic to [0,1) if (and only if) one of the following equivalent conditions holds: (a)X is infinite, Hausdorff, separable, pathwise connected and has exactly one continuous selection; (b)X is infinite, separable, locally connected and has exactly one continuous selection; (c)X is infinite, metric, locally connected and has exactly one continuous selection. Three examples are exhibited which demonstrate the necessity of various assumptions in our results.