论文标题
在各种欧几里得点集
On the variety of Euclidean point sets
论文作者
论文摘要
我们构建了无单胎组件的真实线R的非塑形碎片子空间的连续体。因此，从纯粹的拓扑角度来看，实际线路不仅包含比开放式设置更封闭的集合，而且开放式设置的封闭比开放式集合更多。另一方面，我们表明，如果拓扑观点在度量方向或顺序理论方向上锐化，或者如果r^n替换为dimension n> 1，则这种差异会消失。此外，我们追踪了R的闭合和完全断开的子集的拓扑类型。在此过程中，我们还追踪了单位间隔的无限无限，离散子集的连续类型[0,1]。 （因此，任何无数无限的离散空间都具有非塑形的可衡量压实的连续性。）
We construct a continuum of non-homeomorphic compact subspaces of the real line R without singleton components. Thus from the purely topological point of view the real line contains not only more closed sets than open sets but also more closures of open sets than open sets. On the other hand, we show that this discrepancy vanishes either if the topological point of view is sharpened in the metrical or in the order-theoretical direction, or if R is replaced with R^n for dimension n>1. Furthermore, we track down a continuum of topological types of closed and totally disconnected subsets of R. In doing so we also track down a continuum of metrical types of infinite, discrete subsets of the unit interval [0,1]. (As a consequence, any countably infinite discrete space has a continuum of non-homeomorphic metrizable compactifications.)
