更改

The intuitive concept of dimension of a geometric object X is the number of independent parameters one needs to pick out a unique point inside. However, any point specified by two parameters can be instead specified by one, because the cardinality of the real plane is equal to the cardinality of the real line (this can be seen by an argument involving interweaving the digits of two numbers to yield a single number encoding the same information). The example of a space-filling curve shows that one can even map the real line to the real plane surjectively (taking one real number into a pair of real numbers in a way so that all pairs of numbers are covered) and continuously, so that a one-dimensional object completely fills up a higher-dimensional object.

The intuitive concept of dimension of a geometric object X is the number of independent parameters one needs to pick out a unique point inside. However, any point specified by two parameters can be instead specified by one, because the cardinality of the real plane is equal to the cardinality of the real line (this can be seen by an argument involving interweaving the digits of two numbers to yield a single number encoding the same information). The example of a space-filling curve shows that one can even map the real line to the real plane surjectively (taking one real number into a pair of real numbers in a way so that all pairs of numbers are covered) and continuously, so that a one-dimensional object completely fills up a higher-dimensional object.

几何物体 x 的直观尺寸概念就是一个物体需要多少个独立参数才能找到一个独特的点。但是，任何由两个参数指定的点都可以由一个参数指定，因为实际平面的基数等于实际行的基数(这可以通过一个参数看到，该参数涉及交织两个数字的数字以产生一个编码相同信息的单个数字)。皮亚诺曲线的例子表明，一个人甚至可以完美和连续地映射实际线到真正的平面(把一个实数转换成一对实数，这样所有的数对都被覆盖)，由此一维物体完全填充了一个高维物体。

几何物体X的直观尺寸概念就是一个物体需要多少个独立参数才能找到一个独特的点。但是，任何由两个参数指定的点都可以由一个参数指定，因为实际平面的基数等于实际行的基数(这可以通过一个参数看到，该参数涉及交织两个数字的数字以产生一个编码相同信息的单个数字)。皮亚诺曲线的例子表明，一个人甚至可以完美和连续地映射实际线到真正的平面(把一个实数转换成一对实数，这样所有的数对都被覆盖)，由此一维物体完全填充了一个高维物体。

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。拓扑维度，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 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。拓扑维度，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'' = 1。

But topological dimension is a very crude measure of the local size of a space (size near a point). A curve that is almost space-filling can still have topological dimension one, even if it fills up most of the area of a region. A fractal has an integer topological dimension, but in terms of the amount of space it takes up, it behaves like a higher-dimensional space.

But topological dimension is a very crude measure of the local size of a space (size near a point). A curve that is almost space-filling can still have topological dimension one, even if it fills up most of the area of a region. A fractal has an integer topological dimension, but in terms of the amount of space it takes up, it behaves like a higher-dimensional space.

但是，拓扑维度是对空间局部尺寸(点附近的尺寸)的一个非常粗略的度量。一条几乎是空间填充的曲线仍然可以有一维拓扑，即使它填充了一个区域的大部分面积。分形具有整数的拓扑维数，但就其所占空间的数量而言，它表现得像一个更高维的空间。

但是，拓扑维度是对空间局部尺寸(点附近的尺寸)的一个非常粗略的度量。一条几乎是空间填充的曲线仍然可以有一维拓扑，即使它填充了一个区域的大部分面积。分形具有整数的拓扑维数，但就其所占空间的数量而言，它看起来像一个更高维的空间。

The Hausdorff dimension measures the local size of a space taking into account the distance between points, the metric. Consider the number N(r) of balls of radius at most r required to cover X completely. When r is very small, N(r) grows polynomially with 1/r. For a sufficiently well-behaved  X, the Hausdorff dimension is the unique number d such that N(r) grows as 1/r^d as r approaches zero. More precisely, this defines the box-counting dimension, which equals the Hausdorff dimension when the value d is a critical boundary between growth rates that are insufficient to cover the space, and growth rates that are overabundant.

The Hausdorff dimension measures the local size of a space taking into account the distance between points, the metric. Consider the number N(r) of balls of radius at most r required to cover X completely. When r is very small, N(r) grows polynomially with 1/r. For a sufficiently well-behaved  X, the Hausdorff dimension is the unique number d such that N(r) grows as 1/r^d as r approaches zero. More precisely, this defines the box-counting dimension, which equals the Hausdorff dimension when the value d is a critical boundary between growth rates that are insufficient to cover the space, and growth rates that are overabundant.

豪斯多夫维数测量一个空间的局部大小时，会考虑到点之间距离的度量。考虑半径最大为 r 的球数 n (r) ，需要完全覆盖 x。当 r 很小时，n (r)以1 / r 增长多项式。对于一个表现足够好的 x，豪斯多夫维数是唯一的数 d，这样当 r 趋近于零时 n (r)增长为1 / r sup d / sup。更确切地说，这定义了盒子计数维度，当值 d 是不足以覆盖空间的增长率和过度充裕的增长率之间的关键边界时，它等于豪斯多夫维数。

豪斯多夫维数测量一个空间的局部大小时，会考虑到点之间距离的度量。考虑半径最大为 r 的球数 ''N'' (r) ，需要完全覆盖 ''X''。当 r 很小时，n (r)以1/''r'' 增长多项式。对于一个表现足够好的 ''X''，豪斯多夫维数是唯一的数 d，这样当 r 趋近于零时， N(r)增长为1/r^d 。更确切地说，这定义了盒子计数维度，当值 d 是不足以覆盖空间的增长率和过度充裕的增长率之间的关键边界时，它等于豪斯多夫维数。