在拓撲學和相關數學分支中，一個Hausdorff空間、分離空間或T2空間是一個拓撲空間，其中不同的點有不相交的鄰域。在許多可應用於拓撲空間的分離公理中，「Hausdorff條件」(T2)是最常被使用和討論的。它意味著序列、網和過濾器的極限的唯一性。
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空間是以拓撲的創始人之一Felix Hausdorff的名字命名的。Hausdorff最初對拓撲空間的定義(1914年)將Hausdorff條件作為公理。
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維數是數學家Felix Hausdorff在1918年提出的一個數學概念，它是一組數字的局部大小的度量。，即一個「空間」)，考慮到它的每個成員之間的距離(即，即「空間」中的「點」)。
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").
應用它的數學形式得出:一個點的Hausdorff維數是0，一條線是1，一個正方形是2，一個立方體是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.
也就是說，對於定義了平滑形狀或具有少量角的形狀(傳統幾何和科學的形狀)的點集，Hausdorff維數是一個計數數(整數)，它與拓撲結構對應的維數一致。
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，形式主義也得到了發展，允許計算其他不那麼簡單的對象的維數，其中，僅僅基於它的尺度和自相似性的特性，一個人可以得出結論，特定的對象，包括fractals，具有非整數的Hausdorff維數。
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.
由於Abram Samoilovitch Besicovitch取得了重大的技術進步，允許計算高度不規則集的維數，這個維數通常也被稱為Hausdorff-Besicovitch維數。
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.
豪斯多夫維數是,更具體地說,進一步的維數與一個給定的一組數字,所有成員之間的距離的定義,以及維度的實數,,添加了兩個元素,+∞和∞(解讀為正負無窮,分別)。
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).
提供了豪斯多夫維數的集合稱為擴展的實數,R,和一組數字,所有成員之間的距離稱為定義一個度量空間,以便可以簡練地提出前,說豪斯多夫維數是一個非負實數(R≥0)擴展與任何度量空間。
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.
用數學術語來說，維數概括了實向量空間維數的概念。也就是說，一個n維內積空間的Hausdorff維數等於n，這就構成了前面所說的點的Hausdorff維數為0，線的Hausdorff維數為1等，不規則集合的Hausdorff維數可以是非整數的。
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，前面總結的科赫曲線是由等邊三角形構成的;在每個迭代中,其組件線段分為3段的長度單位,新創建的中間段用作新等邊三角形的底點外,這個基地段然後刪除離開最後一個對象從單位長度的迭代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.
也就是說，在第一次迭代後，每個原始線段都被N=4替換，其中每個自相似副本的長度是原始的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.
換句話說，我們取一個歐幾裡得維數D的物體，在每個方向上將它的線性尺度縮小1/3，使它的長度增加到N=SD。這個方程很容易求出D，得出圖中出現的對數(或自然對數)的比例，並給出——在科赫和其他分形例子中——這些物體的非整數維數。
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維數是較簡單但通常相等的box-counting維數或Minkowski-Bouligand維數的繼承者。
The Hausdorff dimension is a successor to the simpler, but usually equivalent, box-counting or Minkowski–Bouligand dimension.
01豪斯多夫空間 Hausdorff space
定義 Definitions
在拓撲空間x中的點x和點y，如果存在x的鄰域U和y的鄰域V，且U和V不相交(U∩V =)，則可以鄰域將x和y分隔開。
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中所有不同的點都是成對鄰域可分的，則X是一個Hausdorff空間。這個條件是第三個分離公理(在T0和T1之後)，這就是為什麼Hausdorff空間也被稱為T2空間的原因。也使用名稱分隔的空間。
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.
一個相關但較弱的概念是預正則空間。如果任意兩個拓撲可分辨的點可以被鄰域分隔開，則X是一個預正則空間。前分子空間也稱為R1空間。
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.
這兩個條件之間的關係如下。一個拓撲空間是Hausdorff，若且唯若它是pre - egular(即拓撲可區分的點由鄰域分隔開)和Kolmogorov(即不同的點是拓撲可區分的)。若且唯若一個拓撲空間的Kolmogorov商是Hausdorff時，它是預正則的。
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.
等價Equivalences
對於拓撲空間X，以下各項等價:
For a topologicalspaceX, thefollowingareequivalent:
X是Hausdorff空間X is a Hausdorffspace.X中的網的極限是唯一的。Limits of nets in Xareunique.X上的過濾器極限是唯一的Limits of filters on Xareunique.任何單集{x}X 等於x的相交的封閉社區。(一個封閉的社區,x是一個閉集,其中包含一個開集包含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.)對角線Δ= {(x, x) | x∈X}關閉作為產品的一個子集空間X × X。Thediagonal Δ = {(x,x) | x ∈ X} is closed as a subset of theproductspaceX × X.例子和反例 Examples and counterexamples
分析中遇到的空間幾乎都是Hausdorff空間;最重要的是，實數(在實數的標準度量拓撲下)是一個Hausdorff空間。更一般地說，所有的度量空間都是Hausdorff。事實上，許多分析空間，如拓撲群和拓撲流形，在它們的定義中都有明確的Hausdorff條件。
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.
一個簡單的例子是T1但不是Hausdorff的拓撲是定義在無限集上的上有限拓撲。
A simple example of a topology that is T1 but is not Hausdorff is the cofinite topology defined on an infinite set.
偽度量空間通常不是Hausdorff空間，但它們是預估計的，在分析中通常只在Hausdorff規範空間的構建中使用。事實上，當分析人員在一個非Hausdorff空間中運行時，它仍然可能至少是未被發現的，然後他們簡單地用它的Kolmogorov商數來代替它，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.
與此相反，在抽象代數和代數幾何中，非正則空間遇到得更頻繁，特別是在代數變體或環的頻譜上的扎裡斯基拓撲。
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.
它們也出現在直覺邏輯的模型理論中:每一個完整的全集代數都是某個拓撲空間的開集代數，但這個空間不必是預先的，更不必是Hausdorff的，事實上通常兩者都不是。Scott域的相關概念也由非預先定義的空間組成。
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.
當收斂網絡和濾波器存在唯一極限時，表明一個空間是Hausdorff空間，而在非Hausdorff T1空間中，每個收斂序列都有唯一極限。
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.
屬性 Properties
Hausdorff空間的子空間和乘積是Hausdorff空間，但Hausdorff空間的商空間不一定是Hausdorff空間。事實上，每個拓撲空間都可以被實現為某個Hausdorff空間的商。
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空間為T1，意味著所有的單例都是封閉的。類似地，前分子空間是R0。
Hausdorff spaces are T1, meaning that all singletons are closed. Similarly, preregular spaces are R0.
Hausdorff空間的另一個很好的性質是緊集總是閉合的。這可能會失敗在non-Hausdorff Sierpiński空間等空間。
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空間的定義是指點可以被鄰域分隔開。
The definition of a Hausdorff space says that points can be separated by neighborhoods.
事實證明,這意味著這看似強大的東西:在一個緊湊的豪斯多夫空間每一雙也可以由社區,換句話說有一個附近的一套和其他社區,這樣兩個社區是不相交的。這是緊集通常表現為點的一般規則的一個例子。
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.
緊緻條件和預反應性通常意味著更強的分離公理。例如，任何局部緊化的前分子空間都是完全規則的。緊的前分子空間是正規的，這意味著它們滿足Urysohn引理和Tietze擴張定理，並且有服從局部有限開蓋的單位分割。
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.
這些表述的Hausdorff版本是:每個局部緊化的Hausdorff空間都是Tychonoff，每個緊化的Hausdorff空間都是normal Hausdorff。
The Hausdorff versions of these statements are: every locally compact Hausdorff space is Tychonoff, and every compact Hausdorff space is normal Hausdorff.
以下結果是關於映射(連續的和非連續的)到Hausdorff空間的一些技術屬性。
The following results are some technical properties regarding maps (continuous and otherwise) to and from Hausdorff spaces.
設f: X→Y為連續函數，設Y為Hausdorff。那麼f， \{(x,f(x)))\mid x\in x\}的圖就是x×Y的一個閉子集。
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.
設f: X→Y是一個函數，將操作符名{ker} (f)\triangleq \{(X, X ')\mid f(X)=f(X ')}作為X×X的子空間。
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是連續的，Y是Hausdorff，那麼ker(f)是閉的。If f is continuousandY is Hausdorffthenker(f) is closed.如果f是一個開放的服從，ker(f)是封閉的，那麼Y是Hausdorff。If f is an opensurjectionandker(f) is closedthenY is Hausdorff.如果f是一個連續的、開放的滿射(即一個開放商映射)，那麼Y是Hausdorff若且唯若ker(f)是閉的。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是連續映射和Y是豪斯多夫那麼均衡器{\ mbox {eq}} (f, g) = \ {X \ f (X) = g (X)\}是在X上閉合,如果Y是豪斯多夫並且f和g在X的稠密子集上一致那麼f = g。換句話說,連續函數到豪斯多夫空間是由它們的稠密子集上的值決定的。
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是一個閉合餘子，使得f1(Y)對於所有Y∈Y都是緊的，則X為Hausdorff, Y也為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.
設f: X→Y為X的商映射，X為一個緊湊的Hausdorff空間。以下各項是等價的:
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是閉映射。f is a closed map.ker (f)是閉合的。ker(f) is closed.預正則 vs 正則 Preregularity versus regularity
所有的正則空間都是預正則的，就像所有的Hausdorff空間一樣。對於正則空間和Hausdorff空間都成立的拓撲空間有很多結果。
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.
大多數情況下，這些結果適用於所有未出現的空間;它們分別被列在正則空間和Hausdorff空間中，因為預正則空間的概念出現得比較晚。另一方面，那些關於正則性的結果通常也不適用於不正則的Hausdorff空間。
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.
在許多情況下，拓撲空間的另一條件(如旁緊性或局部緊性)如果滿足了預正則性，則意味著正則性。這種情況通常有兩種版本: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空間通常不是正則的，但局部緊化的Hausdorff空間也會是正則的，因為任何Hausdorff空間都是預正則的。
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，從某種角度來看，在這些情況下，真正重要的是預正則性，而不是正則性。然而，定義通常仍然是按照正則性來表述的，因為這種情況比之前的情況更容易理解。
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.
有關此問題的更多信息，請參見分離公理的歷史。
See History of the separation axioms for more on this issue.
變體 Variants
術語「Hausdorff」、「separated」和「preregular」也可以應用於拓撲空間上的變式，如均勻空間、Cauchy空間和收斂空間。在所有這些例子中，將概念統一起來的特徵是，網絡和過濾器的限制(當它們存在時)是惟一的(對於分離的空間)，或者在拓撲不可分辨性(對於前規則空間)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).
結果是，均勻空間，或者更一般的柯西空間，總是前分子的，所以這些情況下的Hausdorff條件簡化為T0條件。
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.
在這些空間中，完整性也是有意義的，在這些情況下，Hausdorffness是完整性的自然伴侶。具體地說，若且唯若每個柯西網至少有一個極限時，一個空間是完備的;若且唯若每個柯西網最多有一個極限時，一個空間是Hausdorff(因為只有柯西網在一開始就有極限)。
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).
代數函數 Algebra of functions
摘要緊Hausdorff空間上連續(實或復)函數的代數是可交換的C*代數，反之，通過Banach-Stone定理可以從連續函數代數的性質中恢復空間的拓撲。
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.
這就引出了非交換幾何，其中非交換C*代數表示非交換空間上的函數代數。
This leads to noncommutative geometry, where one considers noncommutative C*-algebras as representing algebras of functions on a noncommutative space.
學術幽默 Academic humour
在Hausdorff空間中，任意兩點都可以被開集「隔開」，這一雙關語說明了Hausdorff條件。
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曾在波恩大學數學研究所做過研究和演講，在那裡，有一個房間被命名為Hausdorff- raum。這是一個雙關語，因為Raum在德語中的意思是房間和空間。
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豪斯多夫維數 Hausdorff dimension
直覺 Intuition
幾何對象X的維數的直觀概念是一個人需要找出一個唯一的點的獨立參數的數量。
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,任何點指定的兩個參數可以指定的,而不是因為真正的平面的基數等於實線的基數(這可以被一個論點涉及交織兩個數產生一個數字的位數編碼相同的信息)。
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.)
空間填充曲線的例子表明，一個人甚至可以把一個實數變成兩個實數，既可以是滿射的(所以所有的數對都包含在內)，也可以是連續的，這樣一個一維的物體就完全填滿了一個高維的物體。
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.
每個空間填充曲線都會多次命中某些點，且沒有連續的逆。不可能以連續和連續可逆的方式將二維映射到一維。拓撲維，也稱為勒貝格覆蓋維，解釋了為什麼。
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.
這個維數是n如果，在X的每一個小的開球覆蓋中，至少有一個點n + 1個球重疊。例如，當用短的打開間隔覆蓋一條線時，某些點必須覆蓋兩次，從而給出維度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.
但是拓撲維是對空間的局部大小(點附近的大小)的一種非常粗略的度量。幾乎是空間填充的曲線仍然可以有拓撲維度1，即使它填充了一個區域的大部分區域。分形具有整數的拓撲維數，但就其佔用的空間量而言，它的行為類似於高維空間。
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維度衡量的是一個空間的局部大小，同時考慮了點與點之間的距離。考慮半徑最大為r的球的數目N(r)，才能完全覆蓋X。當r很小時，N(r)多項式地以1/r增長。對於一個充分表現良好的X, Hausdorff維數是唯一的數字d，使得N(r)在r趨近於0時以1/rd的速度增長。
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.
更準確地說，這定義了box-counting維數，當d值是不足以覆蓋空間的增長率與過度豐富的增長率之間的臨界邊界時，box-counting維數等於Hausdorff維數。
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.
對於光滑的形狀，或角數較少的形狀，傳統幾何和科學的形狀，Hausdorff維數是一個與拓撲維數一致的整數。但是Benoit Mandelbrot觀察到，在自然界中，具有非整數Hausdorff維的分形集隨處可見。
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.
他發現，對於你周圍看到的大多數粗糙形狀，正確的理想化不是用光滑的理想化形狀，而是用分形的理想化形狀:
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:
雲不是球體，山不是圓錐，海岸線不是圓，樹皮不是光滑的，閃電也不是直線行進。
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.
對於自然界中出現的分形，Hausdorff維數與box-count維數重合。包裝維度是另一個類似的概念，它為許多形狀提供了相同的值，但是在所有這些維度都不同的情況下，有一些記錄良好的例外情況。
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正式定義Formal definitions
豪斯多夫內容Hausdorff content
Let X be a metric space. If S X and d ∈ [0, ∞), the d-dimensional Hausdorff content of S is defined by
設X是度規空間。如果SX和d∈(0,∞),採用分離的內容被定義為
豪斯多夫維數 Hausdorff dimension
X的Hausdorff維數定義如下
TheHausdorffdimensionof X is defined by
同樣的，dimH(X)可以定義為d∈[0，∞)集合的infimum，使得X的d維Hausdorff測度為0。這與d∈[0，∞]集合的上式相同，使得X的d維Hausdorff測度為無窮大(除了後一組數字d為空時，Hausdorff維數為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
可數集的維數為0。Countable sets have Hausdorff dimension 0.歐幾裡得空間n豪斯多夫維數n,而圓S1豪斯多夫維數1。The Euclidean space n has Hausdorff dimension n, and the circle S1 has Hausdorff dimension 1.分形通常是其Hausdorff維數嚴格超過拓撲維數的空間。例如，Cantor集合，一個零維拓撲空間，是它自身的兩個副本的併集，每個副本縮小1/3倍;由此可知，其Hausdorff維數為ln(2)/ln(3)≈0.63。Sierpinski三角形是自身三個副本的併集，每個副本縮小1/2;這就得到了ln(3)/ln(2)≈1.58的Hausdorff維數。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.
空間曲線皮亞諾和Sierpiński曲線有相同的豪斯多夫維數空間填滿。Space-filling curves like the Peano and the Sierpiński curve have the same Hausdorff dimension as the space they fill.二維及以上的布朗運動軌跡幾乎可以確定為二維的Hausdorff運動軌跡。The trajectory of Brownian motion in dimension 2 and above has Hausdorff dimension 2 almost surely.估算大不列顛海岸的Hausdorff維數Estimating the Hausdorff dimension of the coast of Great Britain曼德爾布羅特(Benoit Mandelbrot)早期的一篇論文《英國的海岸線有多長?》(How Long Is the Coast of Britain?)統計自相似和分數維數以及其他作者的後續工作已經表明，許多海岸線的Hausdorff維數是可以估計的。他們的結果從南非海岸線的1.02到英國西海岸的1.25不等。然而，海岸線和許多其他自然現象的「分形維數」在很大程度上是啟發式的，不能嚴格地認為是Hausdorff維數。它們是根據海岸線在大範圍尺度上的尺度特性得出的;然而，它們不包括所有的任意小尺度，其中測量將依賴於原子和亞原子結構，並且沒有很好的定義。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維數的性質 Properties of Hausdorff dimension
Hausdorff維數和歸納維數 Hausdorff dimension and inductive dimension
設X是任意可分離的度量空間。有一個關於X的拓撲概念，它是遞歸定義的。它總是一個整數(或正無窮)，記作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.
假設X是非空的。然後
Suppose X is non-empty. Then
此外，
Moreover,
其中Y在度規空間內同形於X，換句話說，X和Y有相同的基本點集而Y的微分dY在拓撲上等價於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.
這些結果最初是由Edward Szpilrajn(1907-1976)建立的，例如，參見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維數和Minkowski維數 Hausdorff dimension and Minkowski dimension
閔可夫斯基維數與Hausdorff維數相似，至少與Hausdorff維數一樣大，而且在很多情況下它們是相等的。而[0,1]中有理點集的Hausdorff維數為0,Minkowski維數為1。還有緊集，閔可夫斯基維數嚴格大於Hausdorff維數。
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維度與Frostman測度 Hausdorff dimensions and Frostman measures
如果有一個測量μ波萊爾的子集上定義一個度量空間X,以使μ(X) > 0和μ(B (X, r))≤rs適用於某個常數s > 0並且每一個X中的球B(X, r)，那麼dimHaus (X)≥s。Frostman引理提供了一個部分逆命題。
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.
併集和乘積下的行為 Behaviour under unions and products
如果X=\bigcup_{i}X_i是一個有限的或可數的併集，那麼
If X=\bigcup_{i\in I}X_i is a finite or countable union, then
這可以直接從定義中得到驗證。
Thiscan be verifieddirectlyfromthedefinition.
如果X和Y是非空度量空間，則它們乘積的Hausdorff維數滿足
If XandYarenon-emptymetricspaces,thentheHausdorffdimension of theirproductsatisfies
這個不等式是嚴格的。有可能找到兩組維0，它們的乘積維數為1。相反，我們知道當X和Y是Rn的Borel子集時，X×Y的Hausdorff維數從上到下是由X的Hausdorff維數加上Y的上填充維數構成的。這些事實在Mattila(1995)中討論過。
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維數定理 The Hausdorff Dimension Theorem
定理 Theorem.
對於任意給定的{displaystyle r>0，}在n維歐幾裡得空間{\displaystyle R^{n},(n\geq \lceil r\rceil ).}中存在維數為{\displaystyle r}的不可數分形。
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
由自相似條件定義的許多集合具有可以明確確定的維數。大概,一組E是自相似,如果它是一個集值變換的不動點ψ,這是ψ(E) = E,雖然確切的定義如下所示。
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.
假設
Suppose
是Rn上具有壓縮常數rj < 1的壓縮映射。然後有一個唯一的非空緊集
are contractive mappings on Rn with contraction constant rj < 1. Then there is a unique non-empty compact set A such that
該定理由Stefan Banach的壓縮映射不動點定理推廣到具有Hausdorff距離的Rn非空緊子集的完全度量空間。
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.
開集條件 The open set condition
確定自相似集的維數(在某些情況下),我們需要一個技術條件稱為開集條件(OSC)收縮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.
有一個相對緊湊的開集V
There is a relatively compact open set V such that
左邊的併集是兩兩不相交的。
wherethesets in union on theleftarepairwise disjoint.
開集條件分離條件,保證了圖像ψi (V)不重疊「太多」。
Theopensetcondition is a separationconditionthatensurestheimages ψi(V) do notoverlap"toomuch".
定理 Theorem.
假設每個ψi開集條件持有和相似,這是一個組合的等距和擴張周圍。
那麼唯一的不動點ψ是一個集合，它的豪斯多夫維數是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
相似物的收縮係數是膨脹的大小。
The contraction coefficient of a similitude is the magnitude of the dilation.
我們可以用這個定理來計算Sierpinski三角形的Hausdorff維數(有時也稱為Sierpinski墊片)。考慮三個non-collinear點a1, a2, a3 R2在平面上,讓ψi擴張1/2比例的人工智慧。唯一的與ψ映射對應的非空不動點是一個謝爾平斯基鏤墊，並且維數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
對上式兩邊取自然對數，就能求出s，即s = ln(3)/ln(2)Sierpinski墊片是自相似的，滿足OSC要求。一般來說，一個映射不動點集合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
自相似的若且唯若交集
is self-similar if andonly if theintersections
其中s為E的Hausdorff維數，Hs為Hausdorff測度。這一點在Sierpinski墊片的情況下很明顯(交叉點只是點)，但在更普遍的情況下也是正確的:
wheresis theHausdorffdimension of EandHsdenotesHausdorffmeasure.This is clear in thecase of theSierpinskigasket(theintersectionsarejustpoints),but is alsotruemoregenerally:
定理 Theorem.
與前面的定理在同等條件下，ψ的唯一不動點是自相似的。
Underthesameconditions as theprevioustheorem,theuniquefixedpoint of ψ is self-similar.