标题： 空间填充曲线 显示英文标题

标题： Space-Filling Curves

摘要： 我们研究空间填充曲线，它们是从$[0,1]$到某个高维空间的满射连续映射，通常为单位正方形$[0,1]^2$。 特别是，我们定义皮亚诺曲线和勒贝格曲线，并陈述它们的一些性质。 我们还讨论哈恩-马祖尔基维奇定理，该定理描述了$\mathbb{R}^n$中哪些子集是空间填充曲线的像。 最后，我们讨论希尔伯特曲线在现实世界中的应用，特别是谷歌的$S2$单元。 显示英文摘要

摘要： We examine space-filling curves, which are surjective continuous maps from $[0,1]$ to some higher-dimensional space, usually the unit square $[0,1]^2$. In particular, we define Peano's curve and Lebesgue's curve, and state some of their properties. We also discuss the Hahn-Mazurkiewicz theorem, which characterizes those subsets of $\mathbb{R}^n$ that are the image of a space-filling curve. Finally, we discuss real-world applications of Hilbert curves, in particular Google's $S2$ Cells.