数学家传记
莫里斯·弗雷歇是一位法国数学家，在点集拓扑学方面做出了重大贡献，并定义和创立了抽象空间理论。
莫里斯·弗雷歇的父母是Jacques和Zoé 弗雷歇。这是一个新教家庭，在弗雷歇出生时，他的父亲是Maligny一所新教孤儿院的院长。家里有六个孩子，弗雷歇排行第四。当他还是个很小的孩子时，他的父亲Jacques Fréchet被任命为巴黎一所新教学校的校长，全家搬到了那里，对美好的未来抱有很高的期望。然而，政治决定严重损害了弗雷歇家庭舒适的生活。
在弗雷歇出生之前，第三共和国的宪法已经制定。尽管共和派分为两翼，但他们在教会于政治和教育中的角色这一态度上是一致的。其中一位领袖Jules Ferry在1880-85年间担任要职，他引入了新法律，使初等教育免费、义务且世俗化。学校中的宗教教学被“公民教育”取代，从那时起法国教育世俗化了。Jacques Fréchet失去了校长职位并失业。弗雷歇的母亲通过为外国人开办寄宿公寓来为家庭提供经济救助。这带来的好处是弗雷歇在讲外语的人周围长大，并培养了一种国际视野，这种视野将伴随他一生。在法国教育体系从所受法律限制的影响中稳定下来后，Jacques Fréchet能够在新世俗体系中找到一份教学工作。这意味着这个家庭从未享有他们原本可能期望的生活水平，但尽管如此，他们确实恢复了稳定的财务状况。
弗雷歇进入巴黎的布丰中学接受中等教育。在那里，雅克·阿达马教他数学，雅克·阿达马从1890年到1893年在该校任教，之后于1894年被任命为波尔多大学教授。雅克·阿达马立即看到了这个年轻学生的数学潜力，并对他进行个别辅导。在雅克·阿达马搬到波尔多后，这种辅导仍在继续，因为他写信给弗雷歇，给他出数学题，并在有错误时以严厉的批评来批改他的作业。在这种关系中，弗雷歇对他所得到的鼓励和指导极为感激，但他很久以后承认，他一直生活在无法解决所布置问题的恐惧中。
离开学校后，弗雷歇服了兵役，然后在1900年进入巴黎高等师范学校。在那里，他仍然为是专攻物理还是数学而烦恼，最终被说服专攻数学，不是因为他同样不喜欢物理，而是因为进一步学习物理需要他修他不喜欢的化学课程。甚至在1903年获得Agrégation des Sciences Mathematiques之前，弗雷歇就开始发表短论文。到1903年底，他已印刷了四篇论文，其中三篇长达四页。1904年又出现了七篇论文，然后在1905年，他在雅克·阿达马的指导下进行博士研究时，惊人地发表了十一篇论文。与几位在巴黎的美国数学家的接触，特别是埃德温·比德韦尔·威尔逊，导致弗雷歇在美国数学会的出版物上发表了一些早期论文（埃德温·比德韦尔·威尔逊从1903年起是Transactions of the American Mathematical Society的编辑）。弗雷歇大约在这个时候承担的另一项任务是为出版整理埃米尔·博雷尔的讲座。弗雷歇在本科时参加了这些讲座，并在1903-04年冬天整理了这些讲座。Leçons sur les fonctions de variables réelles et les développements en séries de polynômes Ⓣ（关于实变量函数和多项式级数展开的讲座）一书于1905年出版。
弗雷歇于1906年4月2日提交了一篇杰出的博士学位论文Sur quelques points du calcul fonctionnel Ⓣ（关于函数演算的若干问题）。在其中，他引入了度量空间的概念，尽管他没有发明“度量空间”这个名称，这个名称归功于费利克斯·豪斯多夫。这篇论文涉及“函数运算”和“函数演算”，并从雅克·阿达马和维多·沃尔泰拉的思想发展而来。这篇论文的重要性在于，它发展了公理化分析系统，为分析所研究的不同对象提供了一种抽象，类似于群论为代数系统提供抽象的方式。这种类比是弗雷歇自己提出的，他要求他的抽象系统具有足够结构，以便可以研究极限和连续性。他将函数运算定义为定义在任意对象上的数值函数，他希望这些对象包括点、线、函数、数、曲面等。他论文中的函数演算则是对函数运算的系统研究。
作为一位多才多艺的数学家，弗雷歇曾担任贝桑松中学数学教授（1907-08），南特中学数学教授（1908-09），然后担任普瓦捷理学院力学教授（1910-19）。他于1908年与Suzanne Carrive结婚，他们有四个孩子：Hélène、Henri、Denise和Alain。弗雷歇曾安排在1914-15学年到美国厄巴纳的伊利诺伊大学，并已接受那里为期一年的任命。他和家人已经收拾好行李，准备前往港口登船去美国，这时第一次世界大战爆发，弗雷歇需要服兵役。
他于1914年8月4日被动员入伍，但由于他的语言能力——最初是在他母亲经营的外国人机构中获得的——他被分配到英国军队担任翻译。这可能使他的工作比原本稍微安全一些，但尽管如此，他在前线或前线附近度过了大约两年半的时间，因此能幸存下来实属幸运。战争期间，大量法国学者丧生，因为法国人对平等的信念意味着他们倾向于在战壕中作战，而不是从事那些因其专业知识而特别有用的专门战争工作。事实上，有证据表明，弗雷歇曾与一些美国数学家安排，如果他在战争中未能幸存，就出版他的全集。他选择与美国人协商，几乎可以肯定这表明他觉得在那个国家比在自己的国家更受重视。战争期间，他不仅从事这项危险的工作，弗雷歇还继续频繁发表数学论文。人们只能惊叹于他如何在这样的环境下，在几乎没有时间投入数学的情况下，仍能继续从事前沿研究。
在战争期间，弗雷歇保留了他在普瓦捷理学院的职位，尽管无法在那里授课。然而，在战争结束时他从兵役中释放之前，他被选中前往斯特拉斯堡协助重建那里的大学。从1919年到1927年，他同时担任斯特拉斯堡大学高等分析教授和该校数学研究所所长。正如他早年一样，弗雷歇尽管肩负重任，仍能继续产出大量研究成果。他现在承担了重要的行政职责，其中最早的一项是在1920年于斯特拉斯堡筹备和组织国际数学家大会。由于政治原因，这次大会很艰难，因为德国和奥地利数学家被禁止参加，但这导致了强烈的意见和众多争论。他卓越研究成果的一个标志是，在1924年和1925年两年间，他发表了36篇论文。正是在前往斯特拉斯堡之后，他开始对统计学产生兴趣，但在这个阶段他只发表了少量关于probability的文章，他的大部分论文是关于一般分析和拓扑学的。然而，他在斯特拉斯堡教授概率论、统计学和保险数学课程。
从1928年11月起，弗雷歇在巴黎担任职务，但从这时起他更多地专注于统计学。是埃米尔·博雷尔鼓励弗雷歇在巴黎寻求职位，并支持他的候选资格。还有一种说法是，弗雷歇与斯特拉斯堡理学院理事会意见不合，这意味着他既乐于回到巴黎，也不因离开斯特拉斯堡而不快。1928年至1948年退休期间，他在巴黎的数学领域担任过几个不同职务。他曾任高等研究实践学院的研究主任，然后是巴黎理学院的教授。从1929年起，他还担任高等师范学校的分析与力学教授。在[6]中，Armatte详细考察了：-
……弗雷歇从1934年到1936年在国际统计学会发起的反对不当使用相关系数的运动。这场运动采取了不寻常的形式：向全世界同行发送调查问卷，以及在国际统计学会内部的一系列论文、委员会报告和谴责动议。它揭示了在数学统计学最基础的概念之一正发展成为一门自主科学学科的关键时刻，为其提供数学上可靠基础的困难。此外，相关是在多个观察科学领域中广泛用作证明工具的概念。弗雷歇不不屑于通过他关于距离概念的工作为这些数学基础的建设添砖加瓦。而且，由于他尊重应用数学，他使用同样的调查技术——并以同样的斗志——来检验其他统计方法的一致性和相关性，如理论分布参数的估计以及数学在经济和社会问题中的应用。
1942年1月和2月，弗雷歇在葡萄牙讲学。他在里斯本讲授：Les fonctions périodiques, les fonctions presque périodiques et les fonctions assymptotiquement presque périodiques; Applications des fonctions assymptotiquement presque périodiques au théorème ergodique de Birkhoff; Les débuts de la topologie combinatoire; le théorème d'Euler-Cauchy; La théorie des courbes dans les espaces abstraits très généraux; Types homogènes de dimensions Ⓣ（周期函数、殆周期函数和渐近殆周期函数；渐近殆周期函数在Birkhoff遍历定理中的应用；组合拓扑学的开端；莱昂哈德·欧拉-奥古斯丁·路易·柯西定理；非常一般抽象空间中的曲线理论；齐性维数类型）；以及Le développement d'une fonction continue en série de polynômes dans les espaces abstraits Ⓣ（抽象空间中连续函数展开为多项式级数）。此外，他还做了一场面向公众的讲座，题为Les origines des notions mathématiques Ⓣ（数学概念的起源）。
葡萄牙数学学会授予弗雷歇荣誉会员称号，并认可了四项主要成就：
正如我们已经指出的，弗雷歇对点集拓扑学做出了重大贡献，并定义和创立了抽象空间理论。弗雷歇也在统计学、概率论和微积分方面做出了重要贡献。人们对数学进步做出重大贡献的方式各不相同，有些人是通过解决重大问题，另一些人则是通过提出新的研究领域。弗雷歇自己承认他属于后一类。在他1906年的学位论文中，如上所述，他通过对度量空间上的泛函的研究开创了一个全新的领域，并提出了紧性的抽象概念。1907年，他发现了二次昂利·勒贝格可积函数空间上泛函的积分表示定理。里斯独立发现了类似的结果。他引入的general topology所受到的重视程度有所不及，因为1914年费利克斯·豪斯多夫的主要著作出版提供了一个更受欢迎的观点。
弗雷歇最重要的工作包括：
1. Les Espaces abstrait Ⓣ（抽象空间）(1928)，在其中他：-
……为一般拓扑学与抽象分析奠定基础作出了重大贡献。
2. Récherchés théoretiques modernes sur la théorie des probabilités Ⓣ（概率论的现代理论研究）(1937-38)。
3. （与Ky Fan合著）Introduction à la Topologie Combinatoire Ⓣ（组合拓扑学引论）(1946)，关于此书塞缪尔·艾伦伯格写道：-
关于组合拓扑学的有趣读物，数学准备很少的读者也能读懂。讨论的主题有：卡米耶·若尔当曲线定理、地图着色问题、欧拉示性数与曲面的分类。证明非常直观，并不打算完整。包含许多历史注记。
4. Pages choisies d'analyse générale Ⓣ（一般分析选集）(1953)，此书：-
……收录了弗雷歇关于一般分析的论文选段。他还为原始论文补充了简短评论，涉及与其中思想相关的后续发展、对猜想的解答等。这些论文按以下标题分组：Vue d'ensemble；Espaces fonctionnels；Analyse fonctionnelle；Les espaces abstraits；L'analyse générale Ⓣ（概述；函数空间；泛函分析；抽象空间；一般分析）。
5. Les Mathématiques et le concret Ⓣ（数学与具体）（1955），
这是弗雷歇若干论文的合集，按以下总标题分组：Sur les mathématiques en général；Sur le calcul des probabilités et ses applications；Les mathématiciens et la vie. Deux examples。论文因其普遍趣味性和非数学家的可读性而被选出。
最后我们要提到，弗雷歇与当时大多数顶尖数学家都有极为活跃的通信。我们仅记录几个名字：帕维尔·亚历山德罗夫、勒内-路易·贝尔、勒伊岑·布劳威尔、Béla Kerékjártó、卡齐米日·库拉托夫斯基、昂利·勒贝格、保罗·莱维、尼古拉·卢津、P Mahlo、保罗·蒙泰尔、里斯、瓦茨瓦夫·谢尔宾斯基、帕维尔·萨穆伊洛维奇·乌雷松、埃德温·比德韦尔·威尔逊和斯塔尼斯瓦夫·萨伦巴。作为这些通信的一个例子，我们引用[4]，Esther R Phillips对其描述如下：-
据作者所述，弗雷歇与本世纪几乎每一位数学家都有通信。他数量惊人的通信保存在巴黎科学院的档案馆中，为当代数学广泛领域的起源与演变提供了线索。在弗雷歇的手稿中有四十八封信，其中七封由帕维尔·亚历山德罗夫和帕维尔·萨穆伊洛维奇·乌雷松所写，其余（在帕维尔·萨穆伊洛维奇·乌雷松1924年去世后）由帕维尔·亚历山德罗夫所写。与拓扑学发展特别相关的是1920年至1930年代之间所写的信件。在这些最早的信件中，年轻的俄罗斯学者对弗雷歇创立了抽象空间理论表示感谢，他们最早的探究正是以此为基础，并指出他们最初发表作品的来源是尼古拉·卢津在莫斯科大学分析讨论班上提出的问题。这些信件表明，在帕维尔·萨穆伊洛维奇·乌雷松去世后，帕维尔·亚历山德罗夫的探究偏离了“拓扑空间的分类”，变得越来越代数化[组合化]，最终形成他的单纯同调理论。
尽管广受荣誉，但看来弗雷歇在法国国外比在法国国内更受推崇。他受邀在1928年博洛尼亚和1932年奥斯陆的国际数学家大会上作报告。他于1929年当选波兰科学院，1947年当选爱丁堡皇家学会。他还是国际统计学会的成员。他多次在科学院的选举中落选，最终于1956年当选，当时他78岁。
Maurice Fréchet's parents were Jacques and Zoé Fréchet. It was a Protestant family and at the time Maurice was born his father was the director of a Protestant orphanage in Maligny. There were six children in the family, Maurice being the fourth. While he was still a very young child, his father Jacques Fréchet was appointed as head of a Protestant school in Paris and the family moved there with high expectations of a good future. However, political decisions were to severely damage the comfortable life of the Fréchet family.
Before Maurice was born the constitution of the Third Republic had been drawn up. Although the Republicans were divided into two wings, these were united on their attitude to the church's role in politics and education. One of the leaders Jules Ferry held major positions of power during 1880-85 and he brought in new laws to make primary education free, compulsory, and secular. Religious teaching in schools was replaced by "civic education" and from this time on French education was secularised. Jacques Fréchet lost his job as headmaster and was unemployed. Maurice's mother came to the financial rescue of the family by setting up a boarding house for foreigners. This had the benefit that Maurice grew up surrounded by those speaking foreign languages and he developed an international outlook which was to remain with him throughout his life. After the French education system settled down from the impact of the legal restraints put upon it, Jacques Fréchet was able to find a job teaching within the new secular system. It meant that the family never enjoyed the standards they might otherwise have expected, but nevertheless they did return to a stable financial situation.
Maurice entered secondary education at the Lycée Buffon in Paris. There he was taught mathematics by Hadamard who was a teacher at the school from 1890 to 1893 before being appointed professor at the University of Bordeaux in 1894. Hadamard immediately saw the mathematical potential of his young pupil and coached him on an individual basis. This continued after Hadamard moved to Bordeaux, for he wrote to Fréchet setting him mathematical problems, and corrected his work with severe criticisms if there were any errors. The relationship was one in which Fréchet was extremely grateful for the encouragement and guidance that he was receiving, but he admitted much later that he lived in continual fear of not being able to solve the problems he was set.
After leaving school, Fréchet undertook military service before, in 1900, entering the École Normale Supérieure in Paris. There he still worried over the decision on whether to specialise in physics or mathematics, and was eventually persuaded to specialise in mathematics, not because he did not enjoy physics just as much, but rather because further study of physics required him to take chemistry courses which he disliked. Even before he was awarded his Agrégation des Sciences Mathematiques in 1903, Fréchet began publishing short papers. By the end of 1903 he had four papers in print, three of which were four pages long. Seven further papers appeared in 1904, then remarkably eleven papers in 1905 as he undertook research for his doctorate under Hadamard's supervision. Contact with several American mathematicians who were in Paris, in particular Edwin Wilson, led to Fréchet publishing some of his early papers in American Mathematical Society publications (Edwin Wilson was editor of the Transactions of the American Mathematical Society from 1903). Another task undertaken by Fréchet around this time was writing up Borel's lectures for publication. Fréchet attended these lectures while an undergraduate and wrote up the lectures during the winter of 1903-04. The book Leçons sur les fonctions de variables réelles et les développements en séries de polynômes Ⓣ was published in 1905.
Fréchet wrote an outstanding doctoral dissertation Sur quelques points du calcul fonctionnel Ⓣ submitted on 2 April 1906. In it he introduced the concept of a metric space, although he did not invent the name 'metric space' which is due to Hausdorff. The thesis concerns 'functional operations' and 'functional calculus' and is developed from ideas due to Hadamard and Volterra. The importance of the thesis is that it develops axiomatic analysis systems providing an abstraction of different objects studied by analysis in a similar way to group theory providing an abstraction of algebraic systems. This parallel is drawn by Fréchet himself who requires sufficient structure on his abstract systems so that limits and continuity can be studied. He defines a functional operation as a numerically valued function defined on arbitrary objects which he wants to include points, lines, functions, numbers, surfaces etc. The functional calculus of his thesis is then the systematic study of functional operations.
A versatile mathematician, Fréchet served as professor of mathematics at the Lycée in Besançon (1907-08), professor of mathematics at the Lycée in Nantes (1908-09), then professor of mechanics at the Faculty of Science in Poitiers (1910-19). He married Suzanne Carrive in 1908 and they had four children; Hélène, Henri, Denise, and Alain. Fréchet had arranged to spend the academic year 1914-15 at the University of Illinois at Urbana in the United States and had accepted an appointment there for one year. He and his family were packed and ready to travel to the port to board their ship for the United States when World War I broke out and Fréchet was required for military service.
He was mobilised on 4 August 1914 but because of his language skills, initially gained when his mother ran the establishment of foreigners, he was attached to the British Army as an interpreter. This may have resulted in a slightly safer job than he would otherwise have had, but nevertheless he spent about two and a half years at or near the front, so was fortunate to survive. A great many French academics perished during the war, for the French belief in equality meant they tended to fight in the trenches rather than undertake specialised war work for which their expertise made them especially useful. In fact there is evidence that Fréchet had arranged with some American mathematicians to publish his complete works if he did not survive the war. That he chose to negotiate with Americans is almost certainly a sign that he felt more appreciated in that country than in his own. Not only did he undertake this dangerous work during the war, but Fréchet continued to produce frequent mathematics papers. One can only marvel at how he was able to continue with cutting edge research in such circumstances and with so little time to devote to his mathematics.
For the period of the war Fréchet retained his post at the Faculty of Science in Poitiers despite not being able to teach there. However before he was released from military service at the end of the war, he was selected to go to Strasbourg to assist with re-establishing the university there. He was both professor of higher analysis at the University of Strasbourg and Director of the Mathematics Institute there from 1919 to 1927. As he had been in earlier times, Fréchet was able to continue to produce a large research output despite heavy duties. He now had major administrative duties, one of the first being setting up and organising the International Congress of Mathematicians in Strasbourg in 1920. This was a difficult Congress for political reasons, since German and Austrian mathematicians were banned but this resulted in strong opinions and numerous arguments. An indication of his remarkable research output is that he had 36 papers published in the two years 1924 and 1925. It was after going to Strasbourg that he began to become interested in statistics but he only published a small number of articles on probability at this stage, most of his papers being on general analysis and topology. However, he taught courses on probability, statistics, and insurance mathematics at Strasbourg.
From November 1928 Fréchet held posts in Paris, but from this time on he concentrated more on statistics. It was Borel who encouraged Fréchet to seek positions in Paris and he supported his candidacy. There is also a suggestion that Fréchet had a difference of opinion with the Council of the Faculty of Science at Strasbourg which meant he was both pleased to return to Paris and not unhappy at leaving Strasbourg. He held several different positions in the field of mathematics in Paris between 1928 and 1948 when he retired. He was director of studies at the École des Hautes-Études, then professor at the Faculty of Science in Paris. From 1929 he was also professor of analysis and mechanics at the École Normale Supérieure. In [6] Armatte examines in detail the:-
... campaign [Fréchet] sustained from 1934 to 1936 at the International Institute of Statistics against improper uses of the correlation coefficient. This campaign took the unusual form of a survey sent to colleagues all over the world as well as a series of papers, committee reports, and censure motions within the International Institute of Statistics. It sheds light on the difficulties of giving a mathematically sound foundation to one of the most elementary notions of mathematical statistics, at the key moment when it was developing into an autonomous scientific discipline. Moreover, correlation is a notion largely used as an instrument of proof in several domains of observational sciences. Maurice Fréchet did not disdain contributing his own stone to the building of these mathematical foundations via his work on the notion of distance. And, because he respected applied mathematics, he used the same surveying technique - and with the same pugnacity - to examine the consistency and relevance of other statistical methods, like the estimation of the parameters of a theoretical distribution and the application of mathematics to economic and social questions.
In January and February of 1942 Fréchet was lecturing in Portugal. He lectured in Lisbon on: Les fonctions périodiques, les fonctions presque périodiques et les fonctions assymptotiquement presque périodiques; Applications des fonctions assymptotiquement presque périodiques au théorème ergodique de Birkhoff; Les débuts de la topologie combinatoire; le théorème d'Euler-Cauchy; La théorie des courbes dans les espaces abstraits très généraux; Types homogènes de dimensions Ⓣ; and Le développement d'une fonction continue en série de polynômes dans les espaces abstraits Ⓣ. In addition he gave a lecture aimed at the general public on Les origines des notions mathématiques Ⓣ.
The Portuguese Mathematical Society made Fréchet an honorary member and recognised four major achievements:
As we have indicated, Fréchet made major contributions to the topology of point sets, and defined and founded the theory of abstract spaces. Fréchet also made important contributions to statistics, probability and calculus. There are different ways that people make major contributions to the progress of mathematics, some by solving the big questions, others by proposing new areas for research. Fréchet recognised himself that he fell into the latter category. In his dissertation of 1906, discussed above, he started a whole new area with his investigations of functionals on a metric space and formulated the abstract notion of compactness. In 1907 he discovered an integral representation theorem for functionals on the space of quadratic Lebesgue integrable functions. A similar result was discovered independently by Riesz. His introduction of general topology has been somewhat less appreciated than would otherwise have been the case since the publication of Hausdorff's major text in 1914 provided a more popular view.
Fréchet's most important work includes:
1. Les Espaces abstrait Ⓣ (1928), in which he:-
... made a major contribution toward laying the foundations of general topology and abstract analysis.
2. Récherchés théoretiques modernes sur la théorie des probabilités Ⓣ (1937-38).
3. (with Ky Fan) Introduction à la Topologie Combinatoire Ⓣ (1946), about which Eilenberg writes:-
Entertaining reading about combinatorial topology accessible to a reader with very little mathematical preparation. The topics discussed are: the Jordan curve theorem, the map colouring problem, the Euler characteristic and the classification of surfaces. The proofs are very intuitive and are not intended to be complete. Many historical remarks are included.
4. Pages choisies d'analyse générale Ⓣ (1953), which:-
... contains selections by [Fréchet] from his papers on general analysis. He has also supplemented the original papers with short remarks on later developments connected with the ideas in them, answers to conjectures, etc. The papers are grouped under the following headings: Vue d'ensemble; Espaces fonctionnels; Analyse fonctionnelle; Les espaces abstraits; L'analyse générale Ⓣ.
5. Les Mathématiques et le concret Ⓣ (1955),
This is a collection of a number of papers by Fréchet grouped under the general headings: Sur les mathématiques en général; Sur le calcul des probabilités et ses applications; Les mathématiciens et la vie. Deux examples. Papers have been selected for their general interest and readability by non-mathematicians.
Finally let us mention that Fréchet was an extremely active correspondent with most of the leading mathematicians of his day. Let us record just a few of the names: Pavel Sergeevich Aleksandrov, René-Louis Baire, L E J Brouwer, Béla Kerékjártó, Kazimierz Kuratowski, Henri Lebesgue, Paul Lévy, Nikolai Nikolaevich Luzin, P Mahlo, Paul Montel, Frigyes Riesz, Wacław Sierpiński, Pavel Samuilovich Urysohn, Edwin Wilson, and Stanisław Zaremba. As an example of one of these correspondences we refer to [4] which is described by Esther R Phillips as follows:-
Maurice Fréchet, according to the author, corresponded with virtually every mathematician of this century. His prodigious correspondence, preserved in the Archive of the Paris Academy of Sciences, sheds light on the genesis and evolution of a broad body of contemporary mathematics. Among Fréchet's manuscripts are forty-eight letters of which seven were written by P S Aleksandrov and P S Urysohn and the remainder (after Urysohn's death in 1924) by Aleksandrov. Of particular interest in connection with the development of topology are the letters written between 1920 and the 1930s. In the earliest of these letters the young Russian scholars express their gratitude to Fréchet for having created the theory of abstract spaces on which their earliest investigations were based and cite as the source of their first published works problems posed by N Luzin in his analysis seminar at Moscow University. The letters reveal that after Urysohn's death, Aleksandrov's investigations moved away from the "classification of topological spaces" and became increasingly algebraic [combinatorial], culminating in his simplicial theory of homology.
Although widely honoured, it does appear that Fréchet was more highly rated outside France than inside it. He was invited to address the International Congress of Mathematicians in Bologna in 1928 and in Oslo in 1932. He was elected to the Polish Academy of Sciences in 1929 and the Royal Society of Edinburgh in 1947. He was also a member of the International Institute of Statistics. He lost out many times in elections to the Academy of Sciences, eventually being elected in 1956 when he was 78 years old.
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