更改

If <math>X=\bigcup_{i\in I}X_i</math> is a finite or countable union, then

If <math>X=\bigcup_{i\in I}X_i</math> is a finite or countable union, then

如果 math x x i / math 是一个有限或可数的联合，则

如果 <math>X=\bigcup_{i\in I}X_i</math> 是一个有限或可数的联合，则

:<math> \dim_{\operatorname{Haus}}(X) =\sup_{i\in I} \dim_{\operatorname{Haus}}(X_i).</math>

<math> \dim_{\operatorname{Haus}}(X) =\sup_{i\in I} \dim_{\operatorname{Haus}}(X_i).</math>

<math> \dim_{\operatorname{Haus}}(X) =\sup_{i\in I} \dim_{\operatorname{Haus}}(X_i).</math>

<math> \dim_{\operatorname{Haus}}(X) =\sup_{i\in I} \dim_{\operatorname{Haus}}(X_i).</math>

 

This can be verified directly from the definition.

This can be verified directly from the definition.

If X and Y are non-empty metric spaces, then the Hausdorff dimension of their product satisfies

If X and Y are non-empty metric spaces, then the Hausdorff dimension of their product satisfies

如果 x 和 y 是非空度量空间，那么它们乘积的豪斯多夫维数满足

如果 ''X'' 和''Y''是非空度量空间，那么它们乘积的豪斯多夫维数满足

<math> \dim_{\operatorname{Haus}}(X\times Y)\ge \dim_{\operatorname{Haus}}(X)+ \dim_{\operatorname{Haus}}(Y).</math>

<math> \dim_{\operatorname{Haus}}(X\times Y)\ge \dim_{\operatorname{Haus}}(X)+ \dim_{\operatorname{Haus}}(Y).</math>

 

This inequality can be strict. It is possible to find two sets of dimension 0 whose product has dimension 1. In the opposite direction, it is known that when X and Y are Borel subsets of Rⁿ, the Hausdorff dimension of X × Y is bounded from above by the Hausdorff dimension of X plus the upper packing dimension of Y. These facts are discussed in Mattila (1995).

This inequality can be strict. It is possible to find two sets of dimension 0 whose product has dimension 1. In the opposite direction, it is known that when X and Y are Borel subsets of Rⁿ, the Hausdorff dimension of X × Y is bounded from above by the Hausdorff dimension of X plus the upper packing dimension of Y. These facts are discussed in Mattila (1995).

这种不平等可以是绝对的。有可能找到两个维数为0的集合，其乘积的维数为1。在相反的方向上，我们知道当 x 和 y 是 r sup n / sup 的 Borel 子集时，x y 的豪斯多夫维数从上面以 x 的豪斯多夫维数加上 y 的填充维数为界。Mattila (1995)曾就这些情况进行了讨论。

这种不平等可以是绝对的。有可能找到两个维数为0的集合，其乘积的维数为1。相反，我们知道当''X''和''Y''是 '''R'''^''n''的 Borel 子集时， ''X'' × ''Y''的豪斯多夫维数从上面以 ''X''的豪斯多夫维数加上 ''Y''的填充维数为界。Mattila (1995)曾就这些情况进行了讨论。

==Self-similar sets==

==Self-similar sets==