豪斯多夫测度 Hausdorff measure
每天一点纯数学2020-04-27 15:40
在拓扑学和相关数学分支中，一个Hausdorff空间、分离空间或T2空间是一个拓扑空间，其中不同的点有不相交的邻域。在许多可应用于拓扑空间的分离公理中，“Hausdorff条件”(T2)是最常被使用和讨论的。它意味着序列、网和过滤器的极限的唯一性。
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 spaceis a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" (T2) is the most frequently used and discussed. It implies the uniqueness of limits of sequences, nets, and filters.
Hausdorff空间是以拓扑的创始人之一Felix Hausdorff的名字命名的。Hausdorff最初对拓扑空间的定义(1914年)将Hausdorff条件作为公理。
Hausdorff spaces are named after Felix Hausdorff, one of the founders of topology. Hausdorff's original definition of a topological space (in 1914) included the Hausdorff condition as an axiom.
Hausdorff维数是数学家Felix Hausdorff在1918年提出的一个数学概念，它是一组数字的局部大小的度量。，即一个“空间”)，考虑到它的每个成员之间的距离(即，即“空间”中的“点”)。
Hausdorff dimensionis a concept in mathematics introduced in 1918 by mathematician Felix Hausdorff, and it serves as a measure of the local size of a set of numbers (i.e., a "space"), taking into account the distance between each of its members (i.e., the "points" in the "space").
应用它的数学形式得出:一个点的Hausdorff维数是0，一条线是1，一个正方形是2，一个立方体是3。
Applying its mathematical formalisms provides that the Hausdorff dimension of a single point is zero, of a line is 1, and of a square is 2, of a cube is 3.
也就是说，对于定义了平滑形状或具有少量角的形状(传统几何和科学的形状)的点集，Hausdorff维数是一个计数数(整数)，它与拓扑结构对应的维数一致。
That is, for sets of points that define a smooth shape or a shape that has a small number of corners—the shapes of traditional geometry and science—the Hausdorff dimension is a counting number (integer) agreeing with a dimension corresponding to its topology.
然而，形式主义也得到了发展，允许计算其他不那么简单的对象的维数，其中，仅仅基于它的尺度和自相似性的特性，一个人可以得出结论，特定的对象，包括fractals，具有非整数的Hausdorff维数。
However, formalisms have also been developed that allow calculation of the dimension of other less simple objects, where, based solely on its properties of scaling and self-similarity, one is led to the conclusion that particular objects—including fractals—have non-integer Hausdorff dimensions.
由于Abram Samoilovitch Besicovitch取得了重大的技术进步，允许计算高度不规则集的维数，这个维数通常也被称为Hausdorff-Besicovitch维数。
Because of the significant technical advances made by Abram Samoilovitch Besicovitch allowing computation of dimensions for highly irregular sets, this dimension is also commonly referred to as the Hausdorff–Besicovitch dimension.
豪斯多夫维数是,更具体地说,进一步的维数与一个给定的一组数字,所有成员之间的距离的定义,以及维度的实数,,添加了两个元素,+∞和∞(解读为正负无穷,分别)。
The Hausdorff dimension is, more specifically, a further dimensional number associated with a given set of numbers, where the distances between all members of that set are defined, and where the dimension is drawn from the real numbers, , to which two elements have been added, +∞ and ∞ (read as positive and negative infinity, respectively).
提供了豪斯多夫维数的集合称为扩展的实数,R,和一组数字,所有成员之间的距离称为定义一个度量空间,以便可以简练地提出前,说豪斯多夫维数是一个非负实数(R≥0)扩展与任何度量空间。
The set that provides the Hausdorff dimension is called the extended real numbers, R, and a set of numbers where distances between all members are defined is termed a metric space, so that foregoing can be succinctly stated, saying the Hausdorff dimension is a non-negative extended real number (R ≥ 0) associated with any metric space.
用数学术语来说，维数概括了实向量空间维数的概念。也就是说，一个n维内积空间的Hausdorff维数等于n，这就构成了前面所说的点的Hausdorff维数为0，线的Hausdorff维数为1等，不规则集合的Hausdorff维数可以是非整数的。
In mathematical terms, the Hausdorff dimension generalizes the notion of the dimension of a real vector space. That is, the Hausdorff dimension of an n-dimensional inner product space equals n. This underlies the earlier statement that the Hausdorff dimension of a point is zero, of a line is one, etc., and that irregular sets can have noninteger Hausdorff dimensions.
例如，前面总结的科赫曲线是由等边三角形构成的;在每个迭代中,其组件线段分为3段的长度单位,新创建的中间段用作新等边三角形的底点外,这个基地段然后删除离开最后一个对象从单位长度的迭代4。
For instance, the Koch curve summarized earlier is constructed from an equilateral triangle; in each iteration, its component line segments are divided into 3 segments of unit length, the newly created middle segment is used as the base of a new equilateral triangle that points outward, and this base segment is then deleted to leave a final object from the iteration of unit length of 4.
也就是说，在第一次迭代后，每个原始线段都被N=4替换，其中每个自相似副本的长度是原始的1/S = 1/3。
That is, after the first iteration, each original line segment has been replaced with N=4, where each self-similar copy is 1/S = 1/3 as long as the original.
换句话说，我们取一个欧几里得维数D的物体，在每个方向上将它的线性尺度缩小1/3，使它的长度增加到N=SD。这个方程很容易求出D，得出图中出现的对数(或自然对数)的比例，并给出——在科赫和其他分形例子中——这些物体的非整数维数。
Stated another way, we have taken an object with Euclidean dimension, D, and reduced its linear scale by 1/3 in each direction, so that its length increases to N=SD.[4] This equation is easily solved for D, yielding the ratio of logarithms (or natural logarithms) appearing in the figures, and giving—in the Koch and other fractal cases—non-integer dimensions for these objects.
Hausdorff维数是较简单但通常相等的box-counting维数或Minkowski-Bouligand维数的继承者。
The Hausdorff dimension is a successor to the simpler, but usually equivalent, box-counting or Minkowski–Bouligand dimension.
01豪斯多夫空间 Hausdorff space
定义 Definitions
在拓扑空间x中的点x和点y，如果存在x的邻域U和y的邻域V，且U和V不相交(U∩V =)，则可以邻域将x和y分隔开。
Points x and y in a topological space X can be separated by neighbourhoods if there exists a neighbourhood U of x and a neighbourhood V of y such that U and V are disjoint (U ∩ V = ).
如果X中所有不同的点都是成对邻域可分的，则X是一个Hausdorff空间。这个条件是第三个分离公理(在T0和T1之后)，这就是为什么Hausdorff空间也被称为T2空间的原因。也使用名称分隔的空间。
X is a Hausdorff spaceif all distinct points in X are pairwise neighborhood-separable. This condition is the third separation axiom (after T0 and T1), which is why Hausdorff spaces are also called T2 spaces. The name separated space is also used.
一个相关但较弱的概念是预正则空间。如果任意两个拓扑可分辨的点可以被邻域分隔开，则X是一个预正则空间。前分子空间也称为R1空间。
A related, but weaker, notion is that of apreregular space. X is a preregular space if any two topologically distinguishable points can be separated by neighbourhoods. Preregular spaces are also called R1 spaces.
这两个条件之间的关系如下。一个拓扑空间是Hausdorff，当且仅当它是pre - egular(即拓扑可区分的点由邻域分隔开)和Kolmogorov(即不同的点是拓扑可区分的)。当且仅当一个拓扑空间的Kolmogorov商是Hausdorff时，它是预正则的。
The relationship between these two conditions is as follows. A topological space is Hausdorff if and only if it is both preregular (i.e. topologically distinguishable points are separated by neighbourhoods) and Kolmogorov (i.e. distinct points are topologically distinguishable). A topological space is preregular if and only if its Kolmogorov quotient is Hausdorff.
等价Equivalences
对于拓扑空间X，以下各项等价:
For a topologicalspaceX, thefollowingareequivalent:
X是Hausdorff空间X is a Hausdorffspace.X中的网的极限是唯一的。Limits of nets in Xareunique.X上的过滤器极限是唯一的Limits of filters on Xareunique.任何单集{x}X 等于x的相交的封闭社区。(一个封闭的社区,x是一个闭集,其中包含一个开集包含x。)Anysingletonset{x} X is equal to theintersection of all closed neighbourhoods of x. (A closedneighbourhood of x is a closed set thatcontains an opensetcontainingx.)对角线Δ= {(x, x) | x∈X}关闭作为产品的一个子集空间X × X。Thediagonal Δ = {(x,x) | x ∈ X} is closed as a subset of theproductspaceX × X.例子和反例 Examples and counterexamples
分析中遇到的空间几乎都是Hausdorff空间;最重要的是，实数(在实数的标准度量拓扑下)是一个Hausdorff空间。更一般地说，所有的度量空间都是Hausdorff。事实上，许多分析空间，如拓扑群和拓扑流形，在它们的定义中都有明确的Hausdorff条件。
Almost all spaces encountered in analysis are Hausdorff; most importantly, the real numbers (under the standard metric topology on real numbers) are a Hausdorff space. More generally, all metric spaces are Hausdorff. In fact, many spaces of use in analysis, such as topological groups and topological manifolds, have the Hausdorff condition explicitly stated in their definitions.
一个简单的例子是T1但不是Hausdorff的拓扑是定义在无限集上的上有限拓扑。
A simple example of a topology that is T1 but is not Hausdorff is the cofinite topology defined on an infinite set.
伪度量空间通常不是Hausdorff空间，但它们是预估计的，在分析中通常只在Hausdorff规范空间的构建中使用。事实上，当分析人员在一个非Hausdorff空间中运行时，它仍然可能至少是未被发现的，然后他们简单地用它的Kolmogorov商数来代替它，也就是Hausdorff。
Pseudometric spaces typically are not Hausdorff, but they are preregular, and their use in analysis is usually only in the construction of Hausdorff gauge spaces. Indeed, when analysts run across a non-Hausdorff space, it is still probably at least preregular, and then they simply replace it with its Kolmogorov quotient, which is Hausdorff.
与此相反，在抽象代数和代数几何中，非正则空间遇到得更频繁，特别是在代数变体或环的频谱上的扎里斯基拓扑。
In contrast, non-preregular spaces are encountered much more frequently in abstract algebra and algebraic geometry, in particular as the Zariski topology on an algebraic variety or the spectrum of a ring.
它们也出现在直觉逻辑的模型理论中:每一个完整的全集代数都是某个拓扑空间的开集代数，但这个空间不必是预先的，更不必是Hausdorff的，事实上通常两者都不是。Scott域的相关概念也由非预先定义的空间组成。
They also arise in the model theory of intuitionistic logic: every complete Heyting algebra is the algebra of open sets of some topological space, but this space need not be preregular, much less Hausdorff, and in fact usually is neither. The related concept of Scott domain also consists of non-preregular spaces.
当收敛网络和滤波器存在唯一极限时，表明一个空间是Hausdorff空间，而在非Hausdorff T1空间中，每个收敛序列都有唯一极限。
While the existence of unique limits for convergent nets and filters implies that a space is Hausdorff, there are non-Hausdorff T1 spaces in which every convergent sequence has a unique limit.
属性 Properties
Hausdorff空间的子空间和乘积是Hausdorff空间，但Hausdorff空间的商空间不一定是Hausdorff空间。事实上，每个拓扑空间都可以被实现为某个Hausdorff空间的商。
Subspaces and products of Hausdorff spaces are Hausdorff, but quotient spaces of Hausdorff spaces need not be Hausdorff. In fact, every topological space can be realized as the quotient of some Hausdorff space.
Hausdorff空间为T1，意味着所有的单例都是封闭的。类似地，前分子空间是R0。
Hausdorff spaces are T1, meaning that all singletons are closed. Similarly, preregular spaces are R0.
Hausdorff空间的另一个很好的性质是紧集总是闭合的。这可能会失败在non-Hausdorff Sierpiński空间等空间。
Another nice property of Hausdorff spaces is that compact sets are always closed. This may fail in non-Hausdorff spaces such as Sierpiński space.
Hausdorff空间的定义是指点可以被邻域分隔开。
The definition of a Hausdorff space says that points can be separated by neighborhoods.
事实证明,这意味着这看似强大的东西:在一个紧凑的豪斯多夫空间每一双也可以由社区,换句话说有一个附近的一套和其他社区,这样两个社区是不相交的。这是紧集通常表现为点的一般规则的一个例子。
It turns out that this implies something which is seemingly stronger: in a Hausdorff space every pair of disjoint compact sets can also be separated by neighborhoods, in other words there is a neighborhood of one set and a neighborhood of the other, such that the two neighborhoods are disjoint. This is an example of the general rule that compact sets often behave like points.
紧致条件和预反应性通常意味着更强的分离公理。例如，任何局部紧化的前分子空间都是完全规则的。紧的前分子空间是正规的，这意味着它们满足Urysohn引理和Tietze扩张定理，并且有服从局部有限开盖的单位分割。
Compactness conditions together with preregularity often imply stronger separation axioms. For example, any locally compact preregular space is completely regular. Compact preregular spaces are normal, meaning that they satisfy Urysohn's lemma and the Tietze extension theorem and have partitions of unity subordinate to locally finite open covers.
这些表述的Hausdorff版本是:每个局部紧化的Hausdorff空间都是Tychonoff，每个紧化的Hausdorff空间都是normal Hausdorff。
The Hausdorff versions of these statements are: every locally compact Hausdorff space is Tychonoff, and every compact Hausdorff space is normal Hausdorff.
以下结果是关于映射(连续的和非连续的)到Hausdorff空间的一些技术属性。
The following results are some technical properties regarding maps (continuous and otherwise) to and from Hausdorff spaces.
设f: X→Y为连续函数，设Y为Hausdorff。那么f， \{(x,f(x)))\mid x\in x\}的图就是x×Y的一个闭子集。
Let f : X → Y be a continuous function and suppose Y is Hausdorff. Then the graph of f, \{(x,f(x))\mid x\in X\}, is a closed subset of X × Y.
设f: X→Y是一个函数，将操作符名{ker} (f)\triangleq \{(X, X ')\mid f(X)=f(X ')}作为X×X的子空间。
Let f : X → Y be a function and let \operatorname {ker} (f)\triangleq \{(x,x')\mid f(x)=f(x')\} be its kernel regarded as a subspace of X × X.
如果f是连续的，Y是Hausdorff，那么ker(f)是闭的。If f is continuousandY is Hausdorffthenker(f) is closed.如果f是一个开放的服从，ker(f)是封闭的，那么Y是Hausdorff。If f is an opensurjectionandker(f) is closedthenY is Hausdorff.如果f是一个连续的、开放的满射(即一个开放商映射)，那么Y是Hausdorff当且仅当ker(f)是闭的。If f is a continuous,opensurjection(i.e. an openquotientmap)thenY is Hausdorff if and only if ker(f) is closed.如果f, g: X→Y是连续映射和Y是豪斯多夫那么均衡器{\ mbox {eq}} (f, g) = \ {X \ f (X) = g (X)\}是在X上闭合,如果Y是豪斯多夫并且f和g在X的稠密子集上一致那么f = g。换句话说,连续函数到豪斯多夫空间是由它们的稠密子集上的值决定的。
If f,g : X → Y are continuous maps and Y is Hausdorff then the equalizer {\mbox{eq}}(f,g)=\{x\mid f(x)=g(x)\} is closed in X. It follows that if Y is Hausdorff and f and g agree on a dense subset of X then f = g. In other words, continuous functions into Hausdorff spaces are determined by their values on dense subsets.
令f: X→Y是一个闭合余子，使得f1(Y)对于所有Y∈Y都是紧的，则X为Hausdorff, Y也为Hausdorff。
Let f : X → Y be a closed surjection such that f1(y) is compact for all y ∈ Y. Then if X is Hausdorff so is Y.
设f: X→Y为X的商映射，X为一个紧凑的Hausdorff空间。以下各项是等价的:
Let f : X → Y be a quotient map with X a compact Hausdorff space. Then the following are equivalent:
Y是豪斯多夫的。Y is Hausdorff.f是闭映射。f is a closed map.ker (f)是闭合的。ker(f) is closed.预正则 vs 正则 Preregularity versus regularity
所有的正则空间都是预正则的，就像所有的Hausdorff空间一样。对于正则空间和Hausdorff空间都成立的拓扑空间有很多结果。
All regular spaces are preregular, as are all Hausdorff spaces. There are many results for topological spaces that hold for both regular and Hausdorff spaces.
大多数情况下，这些结果适用于所有未出现的空间;它们分别被列在正则空间和Hausdorff空间中，因为预正则空间的概念出现得比较晚。另一方面，那些关于正则性的结果通常也不适用于不正则的Hausdorff空间。
Most of the time, these results hold for all preregular spaces; they were listed for regular and Hausdorff spaces separately because the idea of preregular spaces came later. On the other hand, those results that are truly about regularity generally do not also apply to nonregular Hausdorff spaces.
在许多情况下，拓扑空间的另一条件(如旁紧性或局部紧性)如果满足了预正则性，则意味着正则性。这种情况通常有两种版本:普通版本和Hausdorff版本。
There are many situations where another condition of topological spaces (such as paracompactness or local compactness) will imply regularity if preregularity is satisfied. Such conditions often come in two versions: a regular version and a Hausdorff version.
然Hausdorff空间通常不是正则的，但局部紧化的Hausdorff空间也会是正则的，因为任何Hausdorff空间都是预正则的。
Although Hausdorff spaces are not, in general, regular, a Hausdorff space that is also (say) locally compact will be regular, because any Hausdorff space is preregular.
因此，从某种角度来看，在这些情况下，真正重要的是预正则性，而不是正则性。然而，定义通常仍然是按照正则性来表述的，因为这种情况比之前的情况更容易理解。
Thus from a certain point of view, it is really preregularity, rather than regularity, that matters in these situations. However, definitions are usually still phrased in terms of regularity, since this condition is better known than preregularity.
有关此问题的更多信息，请参见分离公理的历史。
See History of the separation axioms for more on this issue.
变体 Variants
术语“Hausdorff”、“separated”和“preregular”也可以应用于拓扑空间上的变式，如均匀空间、Cauchy空间和收敛空间。在所有这些例子中，将概念统一起来的特征是，网络和过滤器的限制(当它们存在时)是惟一的(对于分离的空间)，或者在拓扑不可分辨性(对于前规则空间)之前是惟一的。
The terms "Hausdorff", "separated", and "preregular" can also be applied to such variants on topological spaces as uniform spaces, Cauchy spaces, and convergence spaces. The characteristic that unites the concept in all of these examples is that limits of nets and filters (when they exist) are unique (for separated spaces) or unique up to topological indistinguishability (for preregular spaces).
结果是，均匀空间，或者更一般的柯西空间，总是前分子的，所以这些情况下的Hausdorff条件简化为T0条件。
As it turns out, uniform spaces, and more generally Cauchy spaces, are always preregular, so the Hausdorff condition in these cases reduces to the T0 condition.
在这些空间中，完整性也是有意义的，在这些情况下，Hausdorffness是完整性的自然伴侣。具体地说，当且仅当每个柯西网至少有一个极限时，一个空间是完备的;当且仅当每个柯西网最多有一个极限时，一个空间是Hausdorff(因为只有柯西网在一开始就有极限)。
These are also the spaces in which completeness makes sense, and Hausdorffness is a natural companion to completeness in these cases. Specifically, a space is complete if and only if every Cauchy net has at least one limit, while a space is Hausdorff if and only if every Cauchy net has at most one limit (since only Cauchy nets can have limits in the first place).
代数函数 Algebra of functions
摘要紧Hausdorff空间上连续(实或复)函数的代数是可交换的C*代数，反之，通过Banach-Stone定理可以从连续函数代数的性质中恢复空间的拓扑。
The algebra of continuous (real or complex) functions on a compact Hausdorff space is a commutative C*-algebra, and conversely by the Banach–Stone theorem one can recover the topology of the space from the algebraic properties of its algebra of continuous functions.
这就引出了非交换几何，其中非交换C*代数表示非交换空间上的函数代数。
This leads to noncommutative geometry, where one considers noncommutative C*-algebras as representing algebras of functions on a noncommutative space.
学术幽默 Academic humour
在Hausdorff空间中，任意两点都可以被开集“隔开”，这一双关语说明了Hausdorff条件。
Hausdorff condition is illustrated by the pun that in Hausdorff spaces any two points can be "housed off" from each other by open sets.
Felix Hausdorff曾在波恩大学数学研究所做过研究和演讲，在那里，有一个房间被命名为Hausdorff- raum。这是一个双关语，因为Raum在德语中的意思是房间和空间。
In the Mathematics Institute of the University of Bonn, in which Felix Hausdorff researched and lectured, there is a certain room designated the Hausdorff-Raum. This is a pun, as Raum means both room and space in German.
02豪斯多夫维数 Hausdorff dimension
直觉 Intuition
几何对象X的维数的直观概念是一个人需要找出一个唯一的点的独立参数的数量。
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.
然而,任何点指定的两个参数可以指定的,而不是因为真正的平面的基数等于实线的基数(这可以被一个论点涉及交织两个数产生一个数字的位数编码相同的信息)。
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.)
空间填充曲线的例子表明，一个人甚至可以把一个实数变成两个实数，既可以是满射的(所以所有的数对都包含在内)，也可以是连续的，这样一个一维的物体就完全填满了一个高维的物体。
The example of a space-filling curve shows that one can even take one real number into two both surjectively (so all pairs of numbers are covered) and continuously, so that a one-dimensional object completely fills up a higher-dimensional object.
每个空间填充曲线都会多次命中某些点，且没有连续的逆。不可能以连续和连续可逆的方式将二维映射到一维。拓扑维，也称为勒贝格覆盖维，解释了为什么。
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.
这个维数是n如果，在X的每一个小的开球覆盖中，至少有一个点n + 1个球重叠。例如，当用短的打开间隔覆盖一条线时，某些点必须覆盖两次，从而给出维度n = 1。
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.
但是拓扑维是对空间的局部大小(点附近的大小)的一种非常粗略的度量。几乎是空间填充的曲线仍然可以有拓扑维度1，即使它填充了一个区域的大部分区域。分形具有整数的拓扑维数，但就其占用的空间量而言，它的行为类似于高维空间。
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.
Hausdorff维度衡量的是一个空间的局部大小，同时考虑了点与点之间的距离。考虑半径最大为r的球的数目N(r)，才能完全覆盖X。当r很小时，N(r)多项式地以1/r增长。对于一个充分表现良好的X, Hausdorff维数是唯一的数字d，使得N(r)在r趋近于0时以1/rd的速度增长。
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/rd as r approaches zero.
更准确地说，这定义了box-counting维数，当d值是不足以覆盖空间的增长率与过度丰富的增长率之间的临界边界时，box-counting维数等于Hausdorff维数。
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.
对于光滑的形状，或角数较少的形状，传统几何和科学的形状，Hausdorff维数是一个与拓扑维数一致的整数。但是Benoit Mandelbrot观察到，在自然界中，具有非整数Hausdorff维的分形集随处可见。
For shapes that are smooth, or shapes with a small number of corners, the shapes of traditional geometry and science, the Hausdorff dimension is an integer agreeing with the topological dimension. But Benot Mandelbrot observed that fractals, sets with noninteger Hausdorff dimensions, are found everywhere in nature.
他发现，对于你周围看到的大多数粗糙形状，正确的理想化不是用光滑的理想化形状，而是用分形的理想化形状:
He observed that the proper idealization of most rough shapes you see around you is not in terms of smooth idealized shapes, but in terms of fractal idealized shapes:
云不是球体，山不是圆锥，海岸线不是圆，树皮不是光滑的，闪电也不是直线行进。
Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.
对于自然界中出现的分形，Hausdorff维数与box-count维数重合。包装维度是另一个类似的概念，它为许多形状提供了相同的值，但是在所有这些维度都不同的情况下，有一些记录良好的例外情况。
For fractals that occur in nature, the Hausdorff and box-counting dimension coincide. The packing dimension is yet another similar notion which gives the same value for many shapes, but there are well documented exceptions where all these dimensions differ.
03正式定义Formal definitions
豪斯多夫内容Hausdorff content
Let X be a metric space. If S X and d ∈ [0, ∞), the d-dimensional Hausdorff content of S is defined by
设X是度规空间。如果SX和d∈(0,∞),采用分离的内容被定义为
豪斯多夫维数 Hausdorff dimension
X的Hausdorff维数定义如下
TheHausdorffdimensionof X is defined by
同样的，dimH(X)可以定义为d∈[0，∞)集合的infimum，使得X的d维Hausdorff测度为0。这与d∈[0，∞]集合的上式相同，使得X的d维Hausdorff测度为无穷大(除了后一组数字d为空时，Hausdorff维数为0)。
Equivalently, dimH(X) may be defined as the infimum of the set of d ∈ [0, ∞) such that the d-dimensional Hausdorff measure of X is zero. This is the same as the supremum of the set of d ∈ [0, ∞) such that the d-dimensional Hausdorff measure of X is infinite (except that when this latter set of numbers d is empty the Hausdorff dimension is zero).
04例子 Examples
可数集的维数为0。Countable sets have Hausdorff dimension 0.欧几里得空间n豪斯多夫维数n,而圆S1豪斯多夫维数1。The Euclidean space n has Hausdorff dimension n, and the circle S1 has Hausdorff dimension 1.分形通常是其Hausdorff维数严格超过拓扑维数的空间。例如，Cantor集合，一个零维拓扑空间，是它自身的两个副本的并集，每个副本缩小1/3倍;由此可知，其Hausdorff维数为ln(2)/ln(3)≈0.63。Sierpinski三角形是自身三个副本的并集，每个副本缩小1/2;这就得到了ln(3)/ln(2)≈1.58的Hausdorff维数。Fractals often are spaces whose Hausdorff dimension strictly exceeds the topological dimension. For example, the Cantor set, a zero-dimensional topological space, is a union of two copies of itself, each copy shrunk by a factor 1/3; hence, it can be shown that its Hausdorff dimension is ln(2)/ln(3) ≈ 0.63.The Sierpinski triangle is a union of three copies of itself, each copy shrunk by a factor of 1/2; this yields a Hausdorff dimension of ln(3)/ln(2) ≈ 1.58.
空间曲线皮亚诺和Sierpiński曲线有相同的豪斯多夫维数空间填满。Space-filling curves like the Peano and the Sierpiński curve have the same Hausdorff dimension as the space they fill.二维及以上的布朗运动轨迹几乎可以确定为二维的Hausdorff运动轨迹。The trajectory of Brownian motion in dimension 2 and above has Hausdorff dimension 2 almost surely.估算大不列颠海岸的Hausdorff维数Estimating the Hausdorff dimension of the coast of Great Britain曼德尔布罗特(Benoit Mandelbrot)早期的一篇论文《英国的海岸线有多长?》(How Long Is the Coast of Britain?)统计自相似和分数维数以及其他作者的后续工作已经表明，许多海岸线的Hausdorff维数是可以估计的。他们的结果从南非海岸线的1.02到英国西海岸的1.25不等。然而，海岸线和许多其他自然现象的“分形维数”在很大程度上是启发式的，不能严格地认为是Hausdorff维数。它们是根据海岸线在大范围尺度上的尺度特性得出的;然而，它们不包括所有的任意小尺度，其中测量将依赖于原子和亚原子结构，并且没有很好的定义。An early paper by Benoit Mandelbrot entitled How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension and subsequent work by other authors have claimed that the Hausdorff dimension of many coastlines can be estimated.Their results have varied from 1.02 for the coastline of South Africa to 1.25 for the west coast of Great Britain. However, 'fractal dimensions' of coastlines and many other natural phenomena are largely heuristic and cannot be regarded rigorously as a Hausdorff dimension.They are based on scaling properties of coastlines at a large range of scales; however, they do not include all arbitrarily small scales, where measurements would depend on atomic and sub-atomic structures, and are not well defined.
05Hausdorff维数的性质 Properties of Hausdorff dimension
Hausdorff维数和归纳维数 Hausdorff dimension and inductive dimension
设X是任意可分离的度量空间。有一个关于X的拓扑概念，它是递归定义的。它总是一个整数(或正无穷)，记作dimind(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 dimind(X).
定理 Theorem.
假设X是非空的。然后
Suppose X is non-empty. Then
此外，
Moreover,
其中Y在度规空间内同形于X，换句话说，X和Y有相同的基本点集而Y的微分dY在拓扑上等价于dX。
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 dY of Y is topologically equivalent to dX.
这些结果最初是由Edward Szpilrajn(1907-1976)建立的，例如，参见Hurewicz和Wallman，第七章。
These results were originally established by Edward Szpilrajn (1907–1976), e.g., see Hurewicz and Wallman, Chapter VII.
Hausdorff维数和Minkowski维数 Hausdorff dimension and Minkowski dimension
闵可夫斯基维数与Hausdorff维数相似，至少与Hausdorff维数一样大，而且在很多情况下它们是相等的。而[0,1]中有理点集的Hausdorff维数为0,Minkowski维数为1。还有紧集，闵可夫斯基维数严格大于Hausdorff维数。
The Minkowski dimension is similar to, and at least as large as, the Hausdorff dimension, and they are equal in many situations. However, the set of rational points in [0, 1] has Hausdorff dimension zero and Minkowski dimension one. There are also compact sets for which the Minkowski dimension is strictly larger than the Hausdorff dimension.
Hausdorff维度与Frostman测度 Hausdorff dimensions and Frostman measures
如果有一个测量μ波莱尔的子集上定义一个度量空间X,以使μ(X) > 0和μ(B (X, r))≤rs适用于某个常数s > 0并且每一个X中的球B(X, r)，那么dimHaus (X)≥s。Frostman引理提供了一个部分逆命题。
If there is a measure μ defined on Borel subsets of a metric space X such that μ(X) > 0 and μ(B(x, r)) ≤ rs holds for some constant s > 0 and for every ball B(x, r) in X, then dimHaus(X) ≥ s. A partial converse is provided by Frostman's lemma.
并集和乘积下的行为 Behaviour under unions and products
如果X=\bigcup_{i}X_i是一个有限的或可数的并集，那么
If X=\bigcup_{i\in I}X_i is a finite or countable union, then
这可以直接从定义中得到验证。
Thiscan be verifieddirectlyfromthedefinition.
如果X和Y是非空度量空间，则它们乘积的Hausdorff维数满足
If XandYarenon-emptymetricspaces,thentheHausdorffdimension of theirproductsatisfies
这个不等式是严格的。有可能找到两组维0，它们的乘积维数为1。相反，我们知道当X和Y是Rn的Borel子集时，X×Y的Hausdorff维数从上到下是由X的Hausdorff维数加上Y的上填充维数构成的。这些事实在Mattila(1995)中讨论过。
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).
Hausdorff维数定理 The Hausdorff Dimension Theorem
定理 Theorem.
对于任意给定的{displaystyle r>0，}在n维欧几里得空间{\displaystyle R^{n},(n\geq \lceil r\rceil ).}中存在维数为{\displaystyle r}的不可数分形。
For any given {\displaystyle r>0,} there are uncountable fractals with Hausdorff dimension {\displaystyle r} in n-dimensional Euclidean space {\displaystyle R^{n},(n\geq \lceil r\rceil ).}
自相似集Self-similar sets
由自相似条件定义的许多集合具有可以明确确定的维数。大概,一组E是自相似,如果它是一个集值变换的不动点ψ,这是ψ(E) = E,虽然确切的定义如下所示。
Many sets defined by a self-similarity condition have dimensions which can be determined explicitly. Roughly, a set E is self-similar if it is the fixed point of a set-valued transformation ψ, that is ψ(E) = E, although the exact definition is given below.
定理 Theorem.
假设
Suppose
是Rn上具有压缩常数rj < 1的压缩映射。然后有一个唯一的非空紧集
are contractive mappings on Rn with contraction constant rj < 1. Then there is a unique non-empty compact set A such that
该定理由Stefan Banach的压缩映射不动点定理推广到具有Hausdorff距离的Rn非空紧子集的完全度量空间。
The theorem follows from Stefan Banach's contractive mapping fixed point theorem applied to the complete metric space of non-empty compact subsets of Rn with the Hausdorff distance.
开集条件 The open set condition
确定自相似集的维数(在某些情况下),我们需要一个技术条件称为开集条件(OSC)收缩ψi序列。
To determine the dimension of the self-similar set A (in certain cases), we need a technical condition called the open set condition (OSC) on the sequence of contractions ψi.
有一个相对紧凑的开集V
There is a relatively compact open set V such that
左边的并集是两两不相交的。
wherethesets in union on theleftarepairwise disjoint.
开集条件分离条件,保证了图像ψi (V)不重叠“太多”。
Theopensetcondition is a separationconditionthatensurestheimages ψi(V) do notoverlap"toomuch".
定理 Theorem.
假设每个ψi开集条件持有和相似,这是一个组合的等距和扩张周围。
那么唯一的不动点ψ是一个集合，它的豪斯多夫维数是s, 而s是以下等式的唯一解
Supposetheopensetconditionholdsandeach ψiis a similitude,that is a composition of an isometry and a dilationaroundsomepoint.
Thentheuniquefixedpoint of ψ is a setwhoseHausdorffdimension is swheresis theuniquesolution of
相似物的收缩系数是膨胀的大小。
The contraction coefficient of a similitude is the magnitude of the dilation.
我们可以用这个定理来计算Sierpinski三角形的Hausdorff维数(有时也称为Sierpinski垫片)。考虑三个non-collinear点a1, a2, a3 R2在平面上,让ψi扩张1/2比例的人工智能。唯一的与ψ映射对应的非空不动点是一个谢尔平斯基镂垫，并且维数s是以下等式的唯一解
We can use this theorem to compute the Hausdorff dimension of the Sierpinski triangle (or sometimes called Sierpinski gasket). Consider three non-collinear points a1, a2, a3 in the plane R2 and let ψi be the dilation of ratio 1/2 around ai. The unique non-empty fixed point of the corresponding mapping ψ is a Sierpinski gasket and the dimension s is the unique solution of
对上式两边取自然对数，就能求出s，即s = ln(3)/ln(2)Sierpinski垫片是自相似的，满足OSC要求。一般来说，一个映射不动点集合E是
Taking natural logarithms of both sides of the above equation, we can solve for s, that is: s = ln(3)/ln(2). The Sierpinski gasket is self-similar and satisfies the OSC. In general a set E which is a fixed point of a mapping
自相似的当且仅当交集
is self-similar if andonly if theintersections
其中s为E的Hausdorff维数，Hs为Hausdorff测度。这一点在Sierpinski垫片的情况下很明显(交叉点只是点)，但在更普遍的情况下也是正确的:
wheresis theHausdorffdimension of EandHsdenotesHausdorffmeasure.This is clear in thecase of theSierpinskigasket(theintersectionsarejustpoints),but is alsotruemoregenerally:
定理 Theorem.
与前面的定理在同等条件下，ψ的唯一不动点是自相似的。
Underthesameconditions as theprevioustheorem,theuniquefixedpoint of ψ is self-similar.
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