**Pavel Urysohn** was a Ukranian mathematician who proved important results in general topology and dimension theory.

- In 1915 Urysohn entered the University of Moscow to study physics and in fact he published his first paper in this year.
- Urysohn graduated in 1919 and continued his studies there working towards his doctorate.
- At this stage Urysohn was interested in analysis, in particular integral equations, and this was the topic of his habilitation.
- Urysohn soon turned to topology.
- 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.
- 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.
- 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.
- Sadly Urysohn had died before even the first part was published.
- Urysohn set out to do far more in this paper than to answer the two questions that Egorov had posed to him.
- 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.
- 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.
- 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).
- They then met Hausdorff who was impressed with Urysohn's results.
- 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.
- He was particularly taken with Urysohn, for whom he developed something like the attachment to a lost son.
- 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 Ⓣ(On the theory of topological spaces) 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.
- He decided to look after the scientific estate of Urysohn as a tribute to the genius of the deceased.

Born 3 February 1898, Odessa, Ukraine. Died 17 August 1924, Batz-sur-Mer, France.

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Origin Ukraine, Topology

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive