论文标题
用点制成的形状拓扑
The Topology of Shapes Made with Points
论文作者
论文摘要
在建筑，城市规划，视觉艺术和其他设计领域中，形状通常由点或基于点集的结构表示制成。用积分制成的形状可以更普遍地理解为形成形状代数$ u_i $的元素（即点）的有限安排，以$ i = 0 $。本文研究了适用于这种形状的拓扑类型。从数学的角度来看，任何“用点制成的形状”等同于有限的空间，因此用点形成的拓扑与有限空间上的拓扑形状没有什么不同：拓扑结构的研究自然与研究点上的预订关系的研究相吻合。确定了这一事实后，讨论了用点的形状拓扑与“无点”图形形状的拓扑之间的一些联系（当$ i> 0 $时），并总结了两者之间的主要区别。
In architecture, city planning, visual arts, and other design areas, shapes are often made with points, or with structural representations based on point-sets. Shapes made with points can be understood more generally as finite arrangements formed with elements (i.e. points) of the algebra of shapes $U_i$, for $i = 0$. This paper examines the kind of topology that is applicable to such shapes. From a mathematical standpoint, any "shape made with points" is equivalent to a finite space, so that topology on a shape made with points is no different than topology on a finite space: the study of topological structure naturally coincides with the study of preorder relations on the points of the shape. After establishing this fact, some connections between the topology of shapes made with points and the topology of "point-free" pictorial shapes (when $i > 0$) are discussed and the main differences between the two are summarized.
