Metric Spaces and Continuous Functions

Abstract

The rethinking process of infinitesimal calculus, that was started with the definition of the limit of a sequence by Bernhard Bolzano (1781–1848) and Augustin-Louis Cauchy (1789–1857) at the beginning of the XIX century and was carried on with the introduction of the system of real numbers by Richard Dedekind (1831–1916) and Georg Cantor (1845–1918) and of the system of complex numbers with the parallel development of the theory of functions by Camille Jordan (1838–1922), Karl Weierstrass (1815– 1897), J. Henri Poincaré (1854–1912), G. F. Bernhard Riemann (1826– 1866), Jacques Hadamard (1865–1963), Emile Borel (1871–1956), René- Louis Baire (1874–1932), Henri Lebesgue (1875–1941) during the whole of the XIX and beginning of the XX century, led to the introduction of new concepts such as open and closed sets, the point of accumulation and the compact set. These notions found their natural collocation and their correct generalization in the notion of a metric space, introduced by Maurice Fréchet (1878–1973) in 1906 and eventually developed by Felix Hausdorff (1869–1942) together with the more general notion of topological space.