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→Hausdorff dimension and inductive dimension
=== Hausdorff dimension and inductive dimension ===
=== Hausdorff dimension and inductive dimension 豪斯多夫维数和归纳维数===
Let ''X'' be an arbitrary [[Separable space|separable]] metric space. There is a [[topology|topological]] notion of [[inductive dimension]] for ''X'' which is defined recursively. It is always an integer (or +∞) and is denoted dim<sub>ind</sub>(''X'').
Let ''X'' be an arbitrary [[Separable space|separable]] metric space. There is a [[topology|topological]] notion of [[inductive dimension]] for ''X'' which is defined recursively. It is always an integer (or +∞) and is denoted dim<sub>ind</sub>(''X'').
Let X be an arbitrary separable metric space. There is a topological notion of inductive dimension for X which is defined recursively. It is always an integer (or +∞) and is denoted dim<sub>ind</sub>(X).
Let X be an arbitrary separable metric space. There is a topological notion of inductive dimension for X which is defined recursively. It is always an integer (or +∞) and is denoted dim<sub>ind</sub>(X).
设 x 是任意可分度量空间。对于 x 有一个递归定义的归纳维数拓扑概念。它总是一个整数(或 + ∞) ，并且表示 dim sub ind / sub (x)。
设''X'' 是[[任意可分度量空间]]。对于''X'' 有一个递归定义的归纳维数拓扑概念。它总是一个整数(或 + ∞) ，并且表示dim<sub>ind</sub>(''X'')。
Theorem. Suppose X is non-empty. Then
Theorem. Suppose X is non-empty. Then
'''定理'''：假设''X'' 是非空的。那么
:<math> \dim_{\mathrm{Haus}}(X) \geq \dim_{\operatorname{ind}}(X). </math>
:<math> \dim_{\mathrm{Haus}}(X) \geq \dim_{\operatorname{ind}}(X). </math>
<math> \dim_{\mathrm{Haus}}(X) \geq \dim_{\operatorname{ind}}(X). </math>
<math> \dim_{\mathrm{Haus}}(X) \geq \dim_{\operatorname{ind}}(X). </math>
Moreover,
Moreover,
此外,
此外,
<math> \inf_Y \dim_{\operatorname{Haus}}(Y) =\dim_{\operatorname{ind}}(X), </math>
<math> \inf_Y \dim_{\operatorname{Haus}}(Y) =\dim_{\operatorname{ind}}(X), </math>
<math> \inf_Y \dim_{\operatorname{Haus}}(Y) =\dim_{\operatorname{ind}}(X), </math>
where ''Y'' ranges over metric spaces [[homeomorphic]] to ''X''. In other words, ''X'' and ''Y'' have the same underlying set of points and the metric ''d''<sub>''Y''</sub> of ''Y'' is topologically equivalent to ''d''<sub>''X''</sub>.
where ''Y'' ranges over metric spaces [[homeomorphic]] to ''X''. In other words, ''X'' and ''Y'' have the same underlying set of points and the metric ''d''<sub>''Y''</sub> of ''Y'' is topologically equivalent to ''d''<sub>''X''</sub>.
where Y ranges over metric spaces homeomorphic to X. In other words, X and Y have the same underlying set of points and the metric d<sub>Y</sub> of Y is topologically equivalent to d<sub>X</sub>.
where Y ranges over metric spaces homeomorphic to X. In other words, X and Y have the same underlying set of points and the metric d<sub>Y</sub> of Y is topologically equivalent to d<sub>X</sub>.
其中 y 是度量空间同胚到 x 的范围。换句话说，x 和 y 具有相同的基本点集，y 的度规 d 子 y / 子拓扑等价于 d 子 x / 子。
其中''Y''是度量空间同胚到 ''X''的范围。换句话说， ''X''和 ''Y''具有相同的基本点集，''Y''的度规''d''<sub>''Y''</sub> 的子拓扑等价于''d''<sub>''X''</sub>。
These results were originally established by Edward Szpilrajn (1907–1976), e.g., see Hurewicz and Wallman, Chapter VII.
These results were originally established by Edward Szpilrajn (1907–1976), e.g., see Hurewicz and Wallman, Chapter VII.
这些结果最初是由 Edward Szpilrajn (1907-1976)建立的，例如，见 Hurewicz 和 Wallman，第七章。
这些结果最初是由 [[Edward Szpilrajn]] (1907–1976)建立的, 参见 Hurewicz and Wallman, Chapter VII.{{full citation needed|date=March 2015}}第七章。
=== Hausdorff dimension and Minkowski dimension 豪斯多夫维数和闵可夫斯基维度===
=== Hausdorff dimension and Minkowski dimension 豪斯多夫维数和闵可夫斯基维度===