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Every space filling curve hits some points multiple times, and does not have a continuous inverse. It is impossible to map two dimensions onto one in a way that is continuous and continuously invertible. The topological dimension, also called Lebesgue covering dimension, explains why. This dimension is n if, in every covering of X by small open balls, there is at least one point where n + 1 balls overlap. For example, when one covers a line with short open intervals, some points must be covered twice, giving dimension n = 1.

Every space filling curve hits some points multiple times, and does not have a continuous inverse. It is impossible to map two dimensions onto one in a way that is continuous and continuously invertible. The topological dimension, also called Lebesgue covering dimension, explains why. This dimension is n if, in every covering of X by small open balls, there is at least one point where n + 1 balls overlap. For example, when one covers a line with short open intervals, some points must be covered twice, giving dimension n = 1.

每条空间填充曲线都会多次撞击某些点，且不存在连续的倒数。将两个维度以连续和连续可逆的方式映射到一个维度是不可能的。'''<font color = '#ff8000'>拓扑维度 topological dimension</font>'''，也被称为拓朴维数，解释了为什么。这个维度是 n，如果在 x 的每个小开球覆盖中，至少有一个点 n + 1个球重叠。例如，当一个点覆盖一条具有短开区间的直线时，某些点必须被覆盖两次，给出维数''n''&nbsp;=&nbsp;1。

每条空间填充曲线都会多次击中某些点，且不存在连续的逆。将二维以连续和连续可逆的方式映射到一维是不可能的。'''<font color = '#ff8000'>拓扑维度 topological dimension</font>'''，也被称为Lebesgue覆盖维数，解释了为什么。如果在 x 的每个小开球覆盖中，至少有一个点 n + 1个球重叠，这个维度是 n。例如，当用短的开区间覆盖一条线时，某些点必须被覆盖两次，给出维数''n''&nbsp;=&nbsp;1。