# Maurice René Fréchet

# Maurice René Fréchet

**1878-1973**

**French Mathematician**

Apioneer in the field of topology, one of the more intellectually challenging disciplines of mathematics, Maurice Fréchet removed the intricacies of topological studies even further from intuitive understanding by introducing an unprecedented degree of abstraction. To mathematicians of the early twentieth century, who were just beginning to accept topology in its more general, concrete form, this development hardly seemed like a step forward. Applications of Fréchet's ideas, however, revealed the value of his methods for solving concrete problems that had longed bedeviled mathematicians.

Fréchet was born on September 10, 1878, in the provincial town of Maligny. His parents were in the business of looking after other people: in Maligny, his father, Jacques, operated an orphanage; and in Paris, to which the family moved soon after Fréchet's birth, his mother, Zoé, ran a boardinghouse that appealed to foreigners. There, Fréchet gained a wide exposure to different cultures, endowing him with the cosmopolitan quality that characterized him—despite the fact that he spent much of his career teaching in the provinces rather than in Paris.

In his high school, or *lycée,* Fréchet had the good fortune to study under a distinguished mathematician, Jacques Hadamard (1865-1963), the author of an important proof regarding the prime-number theorem. Hadamard helped prepare him for the prestigious Ecole Normale Supérieure, which he entered in 1900. Following his graduation, Fréchet went to work with Emile Borel (1871-1956), who, though he was only seven years older than Fréchet, had gained so much experience that he took on the role of a mentor.

In 1908 Fréchet married Suzanne Carrive, with whom he would have four children. He had meanwhile undertaken his doctoral work with Hadamard as his advisor and in 1906 wrote his dissertation on the concept of metric space. This work was a groundbreaking one in topology, which he linked to the set theory of German mathematician Georg Cantor (1845-1918).

At that point, the French mathematical community was just warming up to the more concrete presentation of topology offered by Jules Henri Poincaré (1854-1912). Poincaré had examined the generalization of geometry within the framework of standard Cartesian coordinates, including the *x, y,* and *z* axes. Fréchet, however, took the study of topology into even more rarefied territory, applying it to situations in abstract space, where it was impossible to assign coordinates. In Cartesian space, one could use a distance function based on the Pythagorean theorem, but Fréchet's application required that distance functions be identified in far more generalized terms—what he called "metric spaces."

All of this was a bit too avant-garde for the mathematical establishment of Fréchet's era, though subsequent applications of his ideas proved exceedingly fruitful. Fréchet himself, however, remained at the fringes, teaching in the provinces until 1928, at which point he took a position at the Sorbonne. He remained there until 1949, during which time he saw his ideas become an inspiration to a new generation of mathematicians—only they were Polish, not French. Symbolic of his country's lack of regard for him was the fact that Poland elected him to its academy of sciences in 1929, but France did not give him the same honor until 1956. Awards did come, however, and he eventually received the highly prestigious Legion of Honor Medal. Fréchet, almost 95 years old, died in Paris on June 4, 1973.

**JUDSON KNIGHT**

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**Maurice René Fréchet**