卡齐米日·库拉托夫斯基Kazimierz Kuratowski

卡齐米日·库拉托夫斯基的父亲Marek Kuratowski是华沙的一位著名律师。要了解库拉托夫斯基的求学岁月是什么样的，有必要稍微看一下他出生前后波兰的历史。首先要注意的是，当时波兰实际上并不正式存在。

波兰在1772年被瓜分，南部被称为加利西亚，处于奥地利控制之下。俄罗斯控制了该国其余的大部分地区，在库拉托夫斯基出生前的那些年里，俄罗斯采取了强有力的举措，使所谓的“维斯瓦河地区”受俄罗斯文化支配。在1869年至1874年间实施的一项政策中，所有中学教育都用俄语进行。华沙大学于1869年变成俄罗斯大学之后，华沙才有一所用俄语的大学。然而从1906年起，地下华沙大学建立起来，为那些准备冒着风险在这所非法机构中教与学的人提供波兰语的大学教育。加利西亚虽然处于奥地利控制之下，却保留了波兰文化，并且常常是来自“维斯瓦河地区”的波兰人求学的地方。

库拉托夫斯基九岁时，俄语教育的政策有所放宽，但尽管允许开办波兰语学校，学生仍不能从这样的中学升入大学，除非作为校外考生参加俄语考试。因此，当时“维斯瓦河地区”的大多数波兰人都出国接受大学教育。有些人去了加利西亚，那里虽然处于奥地利控制之下，波兰教育仍然繁荣。然而，库拉托夫斯基在离开中学时决定要成为一名工程师。苏格兰的格拉斯哥大学有一所历史悠久的工程学院，工程学讲席设立于1840年。在库拉托夫斯基看来，它理所当然地是一个学习工程的绝佳之地。

库拉托夫斯基决定去格拉斯哥学习，1913年10月他作为学生在那里注册入学。有趣的是，Sneddon在[15]中讲述道：-

他一定担心自己的名字会给同学带来困难，因为在普通数学班的注册簿上，他的名字写作Casimir Kuratov。

第一学年结束时，库拉托夫斯基获得了数学班级奖。随后他在夏季于技术学院学习化学，并回波兰度假，之后开始第二年的学习。然而，1914年8月回到波兰时正值第一次世界大战爆发，库拉托夫斯基已无法返回苏格兰。尽管他的学业被打断，但对数学而言有一个好处：库拉托夫斯基再也不能学习工程学，而数学将因此获益巨大。

1915年8月，占领波兰多年的俄军撤离华沙。德国和奥匈帝国控制了该国大部分地区，并在华沙设立了一位德国总督。俄军撤离后的首批举措之一就是重建华沙大学，该校于1915年11月作为一所波兰大学开始运作。库拉托夫斯基是大学重新开放后首批学习数学的学生之一。战前他在华沙参加了齐格蒙特·扬尼舍夫斯基和斯特凡·马祖尔凯维奇举办的讨论班。他在[2]中写道：-

早在1917年，[齐格蒙特·扬尼舍夫斯基和斯特凡·马祖尔凯维奇]就在举办一个拓扑学讨论班，这大概是那个新兴而蓬勃发展的领域中的第一个。该讨论班的聚会，在很大程度上伴随着齐格蒙特·扬尼舍夫斯基与斯特凡·马祖尔凯维奇之间有时相当激烈的争论，对参与者来说是一场真正的智力盛宴。

华沙大学还有两位教员也将对库拉托夫斯基产生重大影响。一位是扬·武卡谢维奇，一位研究数理逻辑的哲学教授。第二位是1918年到来的瓦茨瓦夫·谢尔宾斯基。事实上，库拉托夫斯基写的第一篇论文是On the definitions in mathematics，写于1917年，这是他参加扬·武卡谢维奇讨论班时讨论的结果。1919年毕业后，库拉托夫斯基在齐格蒙特·扬尼舍夫斯基和斯特凡·马祖尔凯维奇的指导下攻读博士学位。

1921年，库拉托夫斯基获得了博士学位，但遗憾的是，他的一位导师齐格蒙特·扬尼舍夫斯基已于1920年去世。齐格蒙特·扬尼舍夫斯基是创办新期刊Fundamenta Mathematicae的带头人，该刊第一卷于1920年出版，其中包含一篇由齐格蒙特·扬尼舍夫斯基和库拉托夫斯基合著的论文Sur les continus indécomposableⓉ（论不可分解连续统）。

库拉托夫斯基于1927年被任命为Lwów技术大学的教授。在[4]斯坦尼斯瓦夫·乌拉姆中，这位在库拉托夫斯基开始在Lwów讲学那年才开始大学本科生涯的人写道：-

可以说，他是一位新教授，而我是一名新生。从第一堂课起，我就被他阐述的清晰、逻辑性和精炼以及他所呈现的材料所吸引。……很快我就能回答集合论课程中一些较难的问题，并开始提出其他问题。从一开始，我就感激库拉托夫斯基的耐心和慷慨，愿意花这么多时间与一个新手相处。

Lwów的数学家们在城市的咖啡馆里进行了大量的数学研究。苏格兰咖啡馆最受数学家们普遍欢迎，但库拉托夫斯基却不是，他与胡戈·施泰因豪斯一起（根据斯坦尼斯瓦夫·乌拉姆 [4]）：-

……他通常光顾一家更雅致的茶馆，那里号称拥有波兰最好的糕点。

这家咖啡馆是位于Akademicka街22号的Ludwik Zalewski糖果店。然而，正是在苏格兰咖啡馆里，著名的Scottish Book诞生了，它由在那里工作的数学家们提出的未解决问题组成。库拉托夫斯基（和胡戈·施泰因豪斯）有时会加入同事们在苏格兰咖啡馆的行列，但在数学家们开始将问题记录在《苏格兰书》之前，他已经离开了Lwów。

你可以在THIS LINK看到苏格兰咖啡馆的照片。

然而，在Lwów，库拉托夫斯基与斯特凡·巴拿赫合作，他们解决了一些关于测度论的基本问题。斯坦尼斯瓦夫·乌拉姆，后来成为斯特凡·巴拿赫的研究学生，也与他们合作。正如Arboleda在[1]中所写：-

这是科学合作与理解的一个绝佳范例，也体现了在鼎盛时期组织和鼓励创造性活动的能力。

库拉托夫斯基在利沃夫期间仍与华沙保持联系，每年夏天都会回到首都郊外的家中。1934年，他离开利沃夫，成为华沙大学的数学教授。此后他的职业生涯都在华沙大学度过，尽管他参与的数学活动使他走遍世界各地。正是在这时，库拉托夫斯基开始将精力投入到波兰数学事业中，而不是将所有努力都倾注于自己的研究。然而，他在研究上仍然极为活跃，1936年在普林斯顿度过一个月期间，他与冯·诺伊曼合写了一篇论文。在美国期间，他还接触了罗伯特·李·穆尔的拓扑学小组，结识了此后多年保持联系的数学家们。

齐格蒙特·扬尼舍夫斯基在第一次世界大战结束时撰写的报告中，曾主张波兰数学应集中发展其优势领域。1936年，波兰学术院成立了一个委员会，探讨波兰科学的未来发展方向。库拉托夫斯基成为数学委员会秘书，并于1937年提交了报告。他建议，集中发展优势领域的时代已经过去——这是齐格蒙特·扬尼舍夫斯基提出的——应当在整个数学谱系上全面发展。尤其需要：-

……将应用数学提升到这样的水平，使其能够完成其他科学分支所要求的任务，以及与本国问题相关的任务。

报告建议设立两个研究所，一个从事纯数学，一个从事应用数学，若非政治局势所限，这些建议或许已经实施。1939年德国入侵波兰后，那里的生活变得极其艰难。入侵者有一项策略，旨在终结波兰的知识生活，为此他们将许多学者送往集中营，并杀害了其他人。然而，波兰人有过在这种攻击下生存的经验，他们采用了与俄国统治时期相同的策略，在华沙组织了一所地下大学。库拉托夫斯基冒着生命危险，在整个战争期间在这所非法教育机构中授课。他在[2]中写道：-

我们几乎所有的数学教授都在这些秘密大学授课，当时不少学生如今已成为教授或讲师（Docent）。由于这一地下组织，尽管条件极其艰难，科学工作和教学仍在继续，当然规模要小得多。秘密教育的重要性之一在于维持抵抗精神，以及对未来的乐观和信心，这在占领条件下是如此必要。当时科学家的生活条件确实悲惨。最令人痛苦的是人员的损失。

在两次世界大战之间，波兰在数学教学和研究方面取得了显著飞跃。第二次世界大战结束时，整个教育体系被摧毁，不得不完全重建。正是库拉托夫斯基承担了重建过程中的领导者角色，并通过战后紧接着他担任了八年主席的波兰数学会，着手主张实施他1937年报告中的建议。

这两个研究所——一个纯数学研究所和一个应用数学研究所——被合并为一个单一数学研究所的计划，并于1948年获得批准。库拉托夫斯基于1949年被任命为波兰科学院数学研究所所长。尽管被任命时已53岁，库拉托夫斯基担任这一所长职位达19年。他在波兰科学界还担任其他重要职位。例如，他曾担任波兰科学院的副主席。

库拉托夫斯基在数学出版，特别是波兰数学出版方面也发挥了重要作用。他从1928年起担任Fundamenta Mathematicae编委会成员，1952年接替瓦茨瓦夫·谢尔宾斯基担任主编，并在此后余生一直担任这一职务。他还是重要的Mathematical Monographs丛书的创始人之一和编辑。他为该丛书贡献了第三卷，即他关于拓扑学的专著，我们将在下面再次提到。

作为波兰数学的大使，库拉托夫斯基通过许多国外访问和讲学之旅做出了卓越贡献。他曾在伦敦（1946年）、日内瓦（1948年）、1948-49年期间在美国多所大学、布拉格、柏林、布达佩斯、阿姆斯特丹、罗马、北京（1955年）、广州（1955年）、上海（1955年）、阿格拉（1956年）、勒克瑙（1956年）和孟买（1956年）讲学。所有这一切都发生在旅行受限的斯大林时代，而在旅行变得更容易之后，库拉托夫斯基确实充分利用了机会，多次访问西欧、英国、美国和加拿大。

库拉托夫斯基的主要工作是在拓扑学和集合论领域。他使用极限点的概念给出闭包公理来定义拓扑空间。1922年[1]：-

……他使用布尔代数来独立于点的概念刻画抽象空间的拓扑。后续研究表明，与费利克斯·豪斯多夫用邻域定义拓扑空间一起，闭包算子比基于莫里斯·弗雷歇的收敛（1906）和里斯的聚点（1907）的公理理论产生了更丰富的成果。

库拉托夫斯基的其他主要贡献是在紧性和度量空间方面。他是上文提到的Topologie的作者，这是华沙学派在point set topology方面的最高成就。这部著作的第一卷是几十年来度量空间的主要来源。

他1930年关于非平面图的工作在图论中具有根本重要性，他证明了图$G$可平面的一个充分必要条件是它不包含与$K_{5}$或$K_{3,3}$同胚的子图。

他在集合论方面的工作将函数视为有序对的集合，这使得戈特洛布·弗雷格、查尔斯·桑德斯·皮尔士和恩斯特·施勒德所提出的函数概念成为多余。他还研究了连续统的拓扑学、连通性理论、维数理论，并回答了测度论的问题。

库拉托夫斯基获得了奖项和当选为科学院院士的荣誉。USSR Academy of Sciences、Hungarian Academy、Austrian Academy of Sciences、德意志民主共和国科学院、阿根廷科学院、Accademia dei Lincei、巴勒莫艺术与文学科学院以及爱丁堡皇家学会都选举他为成员。他获得了许多大学的荣誉学位，包括格拉斯哥大学、索邦大学、布拉格大学和弗罗茨瓦夫大学。

斯坦尼斯瓦夫·乌拉姆在他为[2]所写的序言中，用以下的话总结了库拉托夫斯基的贡献：-

库拉托夫斯基不仅作为数学研究中的伟大人物而杰出，而且在于他组织和指导数学研究与教育学派的能力，这在原创科学家中是如此罕见。

Kazimierz Kuratowski's father, Marek Kuratowski was a leading lawyer in Warsaw. To understand what Kuratowski's school years were like it is necessary to look a little at the history of Poland around the time he was born. The first thing to note is that really Poland did not formally exist at this time.

Poland had been partitioned in 1772 and the south was called Galicia and under Austrian control. Russia controlled much of the rest of the country and in the years prior to Kuratowski's birth there had been strong moves by Russia to make "Vistula Land", as it was called, be dominated by Russian culture. In a policy implemented between 1869 and 1874, all secondary schooling was in Russian. Warsaw only had a Russian language university after the University of Warsaw became a Russian university in 1869. From 1906, however, the Underground Warsaw University was set up to provide a Polish university education for those prepared to risk teaching and learning in this illegal institution. Galicia, although under Austrian control, retained Polish culture and was often where Poles from "Vistula Land" went for their education.

When Kuratowski was nine years old the policy of Russian schooling was softened, but although Polish language schools were allowed, a student could not proceed from such a secondary school to university without taking the Russian examinations as an external candidate. As a consequence most Poles in "Vistula Land" at this time went abroad for their university education. Some went to Galicia where, although under Austrian control, Polish education still flourished. Kuratowski, however, when he left secondary school decided that he wanted to become an engineer. The University of Glasgow, in Scotland, had an engineering school with a long established history, the chair of engineering being established in 1840. It rightly appeared to Kuratowski as an outstanding place to study engineering.

After Kuratowski made the decision to study in Glasgow, he matriculated there as a student in October 1913. Interestingly, Sneddon relates in [15]:-

He must have feared that his name would present difficulty to his fellow students for it appears in the registry of the Ordinary Class in Mathematics as Casimir Kuratov.

At the end of his first year Kuratowski was awarded the Class Prize in Mathematics. He then studied chemistry at the Technical College during the summer and returned to Poland for a holiday before starting his second year of study. However, back in Poland in August 1914 at the outbreak of World War I, returning to Scotland became impossible for Kuratowski. Although his education was disrupted, one benefit to mathematics was that Kuratowski could no longer study engineering and mathematics would gain enormously.

In August 1915 the Russian forces which had held Poland for many years withdrew from Warsaw. Germany and Austria-Hungary took control of most of the country and a German governor general was installed in Warsaw. One of the first moves after the Russian withdrawal was the refounding of the University of Warsaw and it began operating as a Polish university in November 1915. Kuratowski was one of the first students to study mathematics when the university reopened. He attended seminars given by Janiszewski and Mazurkiewicz in Warsaw before the end of the war. He writes in [2]:-

As early as 1917 [Janiszewski and Mazurkiewicz] were conducting a topology seminar, presumably the first in that new, exuberantly developing field. The meeting of that seminar, taken up to a large extent with sometimes quite vehement discussions between Janiszewski and Mazurkiewicz, were a real intellectual treat for the participants.

There were two others on the staff at the University of Warsaw who were also to have a major influence on Kuratowski. One was Łukasiewicz, a professor of philosophy who worked on mathematical logic. The second person, who arrived in 1918, was Sierpiński. In fact the first paper which Kuratowski wrote was On the definitions in mathematics, written in 1917, which was a consequence of discussions which he had while attending Łukasiewicz's seminar. After graduating in 1919, Kuratowski undertook his doctoral studies working under Janiszewski and Mazurkiewicz.

In 1921 Kuratowski was awarded his doctorate, but sadly one of his supervisors Janiszewski had died in 1920. Janiszewski had been the leader in a move to set up the new journal Fundamenta Mathematicae and the first volume, which appeared in 1920, contained a joint paper Sur les continus indécomposable Ⓣ by Janiszewski and Kuratowski.

Kuratowski was appointed as a professor at the Technical University of Lwów in 1927. In [4] Ulam, who began his university undergraduate career the year Kuratowski began lecturing in Lwów, wrote:-

He was a freshman professor, so to speak, and I was a freshman student. From the very first lecture I was enchanted by the clarity, logic, and polish of his exposition and the material he presented. ... Soon I could answer some of the more difficult questions in the set theory course, and I began to pose other problems. Right from the start I appreciated Kuratowski's patience and generosity in spending so much time with a novice.

The mathematicians of Lwów did a great deal of mathematical research in the cafés of the city. The Scottish Café was the most popular with the mathematicians in general but not with Kuratowski who, together with Steinhaus (according to Ulam [4]):-

... usually frequented a more genteel tea shop that boasted the best pastry in Poland.

This café was Ludwik Zalewski's Confectionery at 22 Akademicka Street. It was in the Scottish Café, however, that the famous Scottish Book consisting of open questions posed by the mathematicians working there came into being. Kuratowski (and Steinhaus) sometimes joined their colleagues in the Scottish Café but he had left Lwów before the mathematicians began writing down the problems in the Scottish Book.

You can see a picture of the Scottish Café at THIS LINK.

At Lwów, however, Kuratowski worked with Banach and they answered some fundamental problems on measure theory. Ulam, who had become Banach's research student also worked with them. As Arboleda writes in [1]:-

This was a beautiful example of scientific collaboration and understanding, and of the ability to organise and encourage creative activity at its height.

Kuratowski retained his links with Warsaw while in Lwów, returning each summer to his house outside the capital. In 1934 he left Lwów and became professor of mathematics at the University of Warsaw. He was to spend the rest of his career at the University of Warsaw although he became involved in mathematical activities which saw him travelling world-wide. It was now that Kuratowski began to devote his energies to the cause of Polish mathematics rather than to give all his efforts to his research. He was still extremely active in research, however, and while spending a month at Princeton in 1936 he wrote a joint paper with von Neumann. During his time in the United States he also made contact with Robert Moore's topology group, meeting mathematicians whom he would keep in contact with for many years.

Janiszewski had made the case for Polish mathematics concentrating on its areas of strength when he wrote his report at the end of World War I. In 1936 a committee was set up by the Polish Academy of Learning to look at the way forward for Polish science. Kuratowski became secretary to the mathematics committee and his report was made in 1937. He recommended that the time had come to go beyond the era of concentrating on strengths, proposed by Janiszewski, and to develop across the whole of the mathematical spectrum. In particular there was a need:-

... to raise applied mathematics to such a standard that it can fulfil its tasks as required by other branches of science, as well as those tasks connected with the problems of the country.

The recommendations of the report to set up two research institutes, one for pure mathematics and one for applied mathematics, may have been implemented had it not been for the political situation. After the German invasion of Poland in 1939 life there became extremely difficult. There was a strategy by the invaders to put an end to the intellectual life of Poland and to achieve this they sent many academics to concentration camps and murdered others. The Poles had experience of surviving such attacks, however, and they employed the same tactics as they had during the period of Russian domination and organised an underground university in Warsaw. Kuratowski risked his life to teach in this illegal educational establishment through the war. He writes in [2]:-

Almost all our professors of mathematics lectured at these clandestine universities, and quite a few of the students then are now professors or docents themselves. Due to that underground organisation, and in spite of extremely difficult conditions, scientific work and teaching continued, though on a considerably smaller scale of course. The importance of clandestine education consisted among others in keeping up the spirit of resistance, as well as optimism and confidence in the future, which was so necessary in the conditions of occupation. The conditions of a scientist's life at that time were truly tragic. Most painful were the human losses.

Between the two world wars Poland had made a remarkable leap forward in mathematical teaching and research. At the end of World War II the whole educational system was destroyed and had to be completely rebuilt. It was Kuratowski who now took on the role of leader in this rebuilding process and, through the Polish Mathematical Society of which he was president for eight years immediately following the war, he set about arguing for the implementation of the recommendations of his 1937 report.

The two research institutes, one for pure mathematics and one for applied mathematics, were merged into a plan for a single mathematics institute and accepted in 1948. Kuratowski was appointed the Director of the Mathematical Institute of the Polish Academy of Sciences in 1949. Despite being 53 years of age when appointed, Kuratowski held this position of director for 19 years. He held other positions of importance in the Polish scientific scene. For example, he served as a vice president of the Polish Academy of Sciences.

Kuratowski also played a major role in the publishing of mathematics in general and Polish mathematics in particular. He served on the editorial board of Fundamenta Mathematicae from 1928, replacing Sierpiński as editor-in-chief in 1952 and continuing in this role for the rest of his life. He was also one of the founders and an editor of the important Mathematical Monographs series. He contributed the third volume in this series with his monograph on topology which we will mention again below.

As an ambassador for Polish mathematics, Kuratowski did a remarkable job with many foreign visits and lecture tours. He lectured in London (1946), Geneva (1948), many universities in the United States during 1948-49, Prague, Berlin, Budapest, Amsterdam, Rome, Peking (1955), Canton (1955), Shanghai (1955), Agra (1956), Lucknow (1956), and Bombay (1956). All this was during the Stalinist era when travel was restricted, and after travel became easier Kuratowski did indeed take full advantage with many visits to western Europe, Britain, USA, and Canada.

Kuratowski's main work was in the area of topology and set theory. He used the notion of a limit point to give closure axioms to define a topological space. In 1922 [1]:-

... he used Boolean algebra to characterise the topology of an abstract space independently of the notion of points. Subsequent research showed that, together with Felix Hausdorff's definition of topological space in terms of neighbourhoods, the closure operator yielded more fertile results than the axiomatic theories based on Maurice Fréchet's convergence (1906) and Frigyes Riesz's point of accumulation (1907).

Other major contributions by Kuratowski were to compactness and metric spaces. He was the author of Topologie, referred to above, which was the crowning achievement of the Warsaw School in point set topology. The first volume of this work was the major source on metric spaces for several decades.

His 1930 work on non-planar graphs is of fundamental importance in graph theory, he showed that a necessary and sufficient condition for a graph $G$ to be planar is that it does not contain a subgraph homeomorphic to either $K_{5}$ or $K_{3,3}$.

His work in set theory considered a function as a set of ordered pairs and this made the function notion as proposed by Frege, Charles Peirce and Schröder redundant. He also considered the topology of the continuum, the theory of connectivity, dimension theory, and answered measure theory questions.

Kuratowski was honoured with prizes and election to academies. The USSR Academy of Sciences, the Hungarian Academy, the Austrian Academy of Sciences, the Academy of the German Democratic Republic, the Academy of Sciences of Argentina, the Accademia dei Lincei, the Academy of Arts and Letters of Palermo, and the Royal Society of Edinburgh all elected him to membership. He received honorary degrees from many universities including Glasgow, the Sorbonne, Prague and Wrocław.

Ulam, in the preface which he wrote to [2], sums up Kuratowski's contribution in the following words:-

Professor Kuratowski stands out not only as a great figure in mathematical research, but in his ability, so rare among original scientists, to organise and direct schools of mathematical research and education.