帕维尔·萨穆伊洛维奇·乌雷松Pavel Urysohn

帕维尔·萨穆伊洛维奇·乌雷松也被称为乌雷松 Uryson。他的父亲是敖德萨的一位金融家，乌雷松 Samuilovich就出生在这座城市。他出身于一个可追溯到16世纪拉比M Jaffe的家庭。这是一个富裕家庭，乌雷松在莫斯科的一所私立学校接受了中等教育。

1915年，乌雷松进入莫斯科大学学习物理，事实上他在这一年发表了自己的第一篇论文。此时他对物理感兴趣，所以这第一篇论文是关于物理主题的并不奇怪，确实如此，它是关于朱利安·罗威尔·柯立芝管辐射的。然而，他对物理的兴趣很快退居次要地位，因为在莫斯科大学听了尼古拉·卢津和德米特里·叶戈罗夫的讲座之后，他开始专注于数学。

乌雷松于1919年毕业，并继续在那里攻读博士学位。[8]的作者们写道：-

尼古拉·卢津是一位充满活力的数学家，正是他说服乌雷松留在那里，在1919-21年间攻读博士学位。

在这个阶段，乌雷松对分析感兴趣，特别是积分方程，而这正是他教授资格论文（Habilitation）的主题。他于1921年6月获得任教资格，随后成为莫斯科大学的助理教授。

乌雷松很快转向了拓扑学。德米特里·叶戈罗夫向他提出了两个问题，而正是这两个问题占据了他在1921年夏天的时间。德米特里·叶戈罗夫提出的第一个问题是，要找到一个曲线的一般内在拓扑定义，当限制在平面上时，它就变成格奥尔格·康托尔关于平面中无处稠密的连续统的概念。德米特里·叶戈罗夫的第二个问题类似，但应用于曲面，同样要求一个内在拓扑定义。

这些是已经存在了一段时间的困难问题。并不是德米特里·叶戈罗夫提出了新问题，而是他给这位聪明的年轻数学家乌雷松两个真正困难的问题，希望他能提出新的想法。德米特里·叶戈罗夫没有失望，因为乌雷松以极大的决心攻克了这些问题。他没有坐等灵感降临，而是一个接一个地尝试想法，看看是否能给他带来他所寻找的维数的拓扑定义。

与其他年轻的莫斯科数学家一起到布尔科夫村度假，该村位于博尔舍夫镇附近卡利亚兹马河畔，这并没有阻止他尝试寻找维数的“正确”定义。恰恰相反，这对他来说是一个在适宜环境中思考的好机会，而在八月底的一个早晨，他醒来时脑海中有了一个想法，甚至在推敲细节之前他就觉得这是正确的。他立即将他的灵感告诉了他的朋友帕维尔·亚历山德罗夫。

当然，在灵感闪现之后还有大量艰苦的工作。在接下来的一年里，乌雷松通过推导后果，在拓扑学中建立了一个全新的维数理论领域。对莫斯科的拓扑学家来说，这是一个激动人心的时期，因为乌雷松讲授连续统的拓扑学，他最新的结果常常在证明后不久就在课程中呈现。他在1922年期间发表了一系列关于这个主题的简短笔记。完整的理论发表在一篇文章中，昂利·勒贝格接受了该文章在巴黎的科学院的Comptes rendus上发表。这给了乌雷松一个国际平台来展示他的想法，立即引起了像大卫·希尔伯特这样的数学家的兴趣。

乌雷松在Fundamenta mathematicae中发表了他的维数理论的完整版本。他在1923年写了两部分的重要论文，但直到1925年和1926年才付印。遗憾的是，乌雷松在第一部分出版之前就已经去世了。论文以乌雷松陈述他的目标开始，即：-

为了指出仍然值得称为“线”和“面”的最一般的集合……

事实上，乌雷松在这篇论文中着手做的远不止回答德米特里·叶戈罗夫向他提出的那两个问题。正如Crilly和Johnson所写[8]：-

他不仅寻求曲线和曲面的定义，还寻求$n$维Cantor流形的定义，从而也寻求维数本身的定义。事实上，维数概念是他关注的中心。

乌雷松在详细制定其拓扑维数理论时并不知道勒伊岑·布劳威尔的贡献，勒伊岑·布劳威尔实际上已于1913年就这一主题发表过文章。然而，他给出的是一个整体定义，这与乌雷松的局部维数定义形成对比。乌雷松思想的另一个重要方面是，他是在紧度量空间的背景下提出这些思想的。乌雷松去世后，帕维尔·亚历山德罗夫论证说，尽管乌雷松的维数定义是针对度量空间给出的，但它 nevertheless 与卡尔·门格尔为一般拓扑空间给出的定义完全等价。

乌雷松于1923年访问了哥廷根。他给哥廷根数学会的报告引起了大卫·希尔伯特的兴趣，而在哥廷根期间，他了解到了勒伊岑·布劳威尔在我们上面提到的1913年论文中对该领域的贡献。乌雷松在哥廷根研究勒伊岑·布劳威尔的论文时发现了其中关于维数定义的一个错误，并轻松构造了一个反例。他在马尔堡德国数学会年会上遇到了勒伊岑·布劳威尔，两人都做了演讲，乌雷松在他的演讲中提到了勒伊岑·布劳威尔的错误及其反例。这是一个让勒伊岑·布劳威尔开始重新思考拓扑学的场合，因为他的兴趣已经转向直觉主义，这也是他在马尔堡演讲的主题。

1924年夏天，乌雷松再次与帕维尔·亚历山德罗夫一起出发，前往欧洲旅行，途经德国、荷兰和法国。两位数学家再次拜访了大卫·希尔伯特，到5月7日他们一定已经离开了，因为大卫·希尔伯特在那天写信给乌雷松，告诉他他与帕维尔·亚历山德罗夫合写的论文已被Mathematische Annalen接受发表（见下文）。这封信收录在[11]中，还感谢乌雷松送给大卫·希尔伯特的鱼子酱，并表示希望乌雷松明年夏天再来访问。

他们随后会见了费利克斯·豪斯多夫，后者对乌雷松的结果印象深刻。他还写了一封信给乌雷松，日期为1924年8月11日（见[11]）。这封信讨论了乌雷松的度量化定理及其通用可分度量空间的构造。构造一个包含任意度量空间的等距像的通用度量空间，是乌雷松最后的结果之一。像大卫·希尔伯特一样，费利克斯·豪斯多夫也表示希望乌雷松明年夏天再次来访。Van Dalen在[13]中写到了他们最后一次数学访问，那次访问是去勒伊岑·布劳威尔：-

这一次[乌雷松和帕维尔·亚历山德罗夫]拜访了勒伊岑·布劳威尔，后者对这两位俄罗斯人印象极为良好。他尤其喜欢乌雷松，对他产生了一种类似对失散儿子的依恋之情。

这次访问之后，两位数学家继续他们的假期，前往布列塔尼，在那里租了一间小屋。乌雷松在一次例行的离岸游泳中，于波涛汹涌的海面上溺水身亡。

乌雷松不仅是帕维尔·亚历山德罗夫的“不可分离的朋友”，两人还合作发表了重要著作，如Zur Theorie der topologischen RäumeⓉ（《拓扑空间论》），于1924年发表在Mathematische Annalen上。除了上面讨论的维数理论之外，乌雷松的主要贡献还包括引入并研究了一类正规曲面、度量化定理，以及一个关于将任意赋范空间映射到具有可数基的希尔伯特空间中的重要存在性定理。他尤其因“乌雷松引理”而被人们铭记，该引理证明了存在某个连续函数，在特定的闭子集上取值为0和1。

乌雷松的去世勒伊岑·布劳威尔和帕维尔·亚历山德罗夫确保了他留下的数学成果得到妥善处理。正如van Dalen所写[13]：-

勒伊岑·布劳威尔悲痛欲绝。他决定整理乌雷松的科学遗产，以此向逝者的天才致敬。他与帕维尔·亚历山德罗夫一起完成了这项任务。

Crilly和Johnson写道[8]：-

考虑到他只有三年时间致力于拓扑学，他以才华和热情在自己选择的领域留下了印记。他将这门学科转变成了现代数学的一个丰富领域。如果他没有如此年轻就去世，他本可以取得多大的成就？

Pavel Urysohn is also known as Pavel Uryson. His father was a financier in Odessa, the town in which Pavel Samuilovich was born. He came from a family descended from the sixteenth century Rabbi M Jaffe. It was a well-off family and Urysohn received his secondary education in Moscow at a private school there.

In 1915 Urysohn entered the University of Moscow to study physics and in fact he published his first paper in this year. Being interested in physics at this time it is not surprising that this first paper was on a physics topic, and indeed it was, being on Coolidge tube radiation. However his interest in physics soon took second place for after attending lectures by Luzin and Egorov at the University of Moscow he began to concentrate on mathematics.

Urysohn graduated in 1919 and continued his studies there working towards his doctorate. The authors of [8] write:-

Luzin was a dynamic mathematician and it was he who persuaded Urysohn to stay on in order to study for a doctorate during 1919-21.

At this stage Urysohn was interested in analysis, in particular integral equations, and this was the topic of his habilitation. He was awarded his habilitation in June 1921 and, following this, became an assistant professor at the University of Moscow.

Urysohn soon turned to topology. He was asked two questions by Egorov and it was these which occupied him during the summer of 1921. The first question that Egorov posed was to find a general intrinsic topological definition of a curve which when restricted to the plane became Cantor's notion of a continuum which is nowhere dense in the plane. The second of Egorov's questions was a similar one but applied to surfaces, again asking for an intrinsic topological definition.

These were difficult questions which had been around for some time. It was not that Egorov had come up with new questions, rather he was giving the bright young mathematician Urysohn two really difficult problems in the hope that he might come up with new ideas. Egorov was not to be disappointed, for Urysohn attacked the questions with great determination. He did not sit still waiting for inspiration to strike, rather he tried one idea after another to see if it would give him the topological definition of dimension that he was looking for.

A holiday with other young Moscow mathematicians to the village of Burkov, on the banks of the river Kalyazmy near to the town of Bolshev, did not stop him trying to find the "right" definition of dimension. Quite the opposite, it was a good chance for him to think in congenial surroundings, and one morning near the end of August he woke up with an idea in his mind which he felt, even before working through the details, was right. Immediately he told his friend Aleksandrov about his inspiration.

Of course there was a lot of hard work after the moment of inspiration. During the following year Urysohn worked through the consequences building a whole new area of dimension theory in topology. It was an exciting time for the topologists in Moscow for Urysohn lectured on the topology of continua and often his latest results were presented in the course shortly after he had proved them. He published a series of short notes on this topic during 1922. The complete theory was presented in an article which Lebesgue accepted for publication in the Comptes rendus of the Academy of Sciences in Paris. This gave Urysohn an international platform for his ideas which immediately attracted the interest of mathematicians such as Hilbert.

Urysohn published a full version of his dimension theory in Fundamenta mathematicae. He wrote a major paper in two parts in 1923 but they did not appear in print until 1925 and 1926. Sadly Urysohn had died before even the first part was published. The paper begins with Urysohn stating his aim which was:-

To indicate the most general sets that still merit being called "lines" and "surfaces" ...

In fact Urysohn set out to do far more in this paper than to answer the two questions that Egorov had posed to him. As Crilly and Johnson write [8]:-

Not only did he seek definitions of curve and surface, but also definitions of $n$-dimensional Cantorian manifold and hence of dimension itself. The dimension concept was, in fact, the centre of his attention.

Although Urysohn did not know of Brouwer's contribution when he worked out the details of his theory of topological dimension, Brouwer had in fact published on that topic in 1913. He had given a global definition, however, and this was in contrast to Urysohn's local definition of dimension. Another important aspect of Urysohn's ideas was the fact that he presented them in the context of compact metric spaces. After Urysohn's death, Aleksandrov argued that although Urysohn's definition of dimension was given for a metric space, it is, nevertheless, completely equivalent to the definition given by Menger for general topological spaces.

Urysohn visited Göttingen in 1923. His reports to the Mathematical Society of Göttingen interested Hilbert and while in Göttingen he learnt of Brouwer's contributions to the area made in the paper of 1913 to which we referred above. Urysohn spotted an error in Brouwer's paper regarding a definition of dimension while he was studying it in Göttingen and easily constructed a counter-example. He met Brouwer at the annual meeting of the German Mathematical Society in Marburg where both gave lectures and Urysohn mentioned Brouwer's error, and his counter-example, in his talk. It was an occasion which made Brouwer begin to think about topology again, for his interests had turned to intuitionism, the subject of his talk at Marburg.

In the summer of 1924 Urysohn set off again with Aleksandrov on a European trip through Germany, Holland and France. Again the two mathematicians visited Hilbert and, by 7 May, they must have left since Hilbert wrote to Urysohn on that day telling him his paper with Aleksandrov was accepted for publication in Mathematische Annalen (see below). This letter, given in [11], also thanks Urysohn for caviar he had given Hilbert, and expresses the hope that Urysohn will visit again the following summer.

They then met Hausdorff who was impressed with Urysohn's results. He also wrote a letter to Urysohn which was dated 11 August 1924 (see [11]). The letter discusses Urysohn's metrization theorem and his construction of a universal separable metric space. The construction of a universal metric space, containing an isometric image of any metric space, was one of Urysohn's last results. Like Hilbert, Hausdorff expressed the hope that Urysohn would visit again the following summer. Van Dalen writes in [13] about their final mathematical visit which was to Brouwer:-

This time [Urysohn and Aleksandrov] visited Brouwer, who was most favourably impressed by the two Russians. He was particularly taken with Urysohn, for whom he developed something like the attachment to a lost son.

After this visit the two mathematicians continued their holiday to Brittany where they rented a cottage. Urysohn drowned in rough seas while on one of their regular swims off the coast.

Urysohn was not only an "inseparable friend" to Aleksandrov but the two collaborated on important publications such as Zur Theorie der topologischen Räume Ⓣ published in Mathematische Annalen in 1924. Urysohn's main contributions, in addition to the theory of dimension discussed above, are the introduction and investigation of a class of normal surfaces, metrization theorems, and an important existence theorem concerning mapping an arbitrary normed space into a Hilbert space with countable basis. He is remembered particularly for 'Urysohn's lemma' which proves the existence of a certain continuous function taking values 0 and 1 on particular closed subsets.

After Urysohn's death Brouwer and Aleksandrov made sure that the mathematics he left was properly dealt with. As van Dalen writes [13]:-

Brouwer was broken hearted. He decided to look after the scientific estate of Urysohn as a tribute to the genius of the deceased. Together with Aleksandrov he acquitted himself of this task.

Crilly and Johnson write [8]:-

Considering that he only had three years to devote to topology, he made his mark in his chosen field with brilliance and passion. He transformed the subject into a rich domain of modern mathematics. How much more might he there have been, had he not died so young?