linguistic:translation, structural:summarization
History

Arthur Cayley, in his article "On Distance", extended metric concepts beyond Euclidean geometry into domains bounded by a conic in a projective space. His distance was given by logarithm of a cross ratio. Any projectivity leaving the conic stable also leaves the cross ratio constant, so isometries are implicit. This method provides models for elliptic geometry and hyperbolic geometry, and Felix Klein, in several publications, established the field of non-euclidean geometry through the use of the Cayley-Klein metric.

The idea of an abstract space with metric properties was addressed in 1906 by René Maurice Fréchet and the term metric space was coined by Felix Hausdorff in 1914.

Fréchet's work laid the foundation for understanding convergence, continuity, and other key concepts in non-geometric spaces. This allowed mathematicians to study functions and sequences in a broader and more flexible way. This was important for the growing field of functional analysis. Mathematicians like Hausdorff and Stefan Banach further refined and expanded the framework of metric spaces. Hausdorff introduced topological spaces as a generalization of metric spaces. Banach's work in functional analysis heavily relied on the metric structure. Over time, metric spaces became a central part of modern mathematics. They have influenced various fields including topology, geometry, and applied mathematics. Metric spaces continue to play a crucial role in the study of abstract mathematical concepts.