Homeomorphism

Part of speech: noun

Definitions

1. A relationship characterized by a continuous, bijective mapping connecting two topological structures exists, preserving their essential properties and allowing transformations without alteration
2. Such a correspondence between two topological spaces is defined by a continuous one-to-one and onto function, along with a continuous inverse that maintains the integrity of their structures
3. A continuous, invertible mapping that links two spaces in a way that preserves topological properties without tearing or gluing

Etymology: "Homeomorphism" is a mathematical coinage built from two Greek-derived parts: "homeo-", from Greek "homoios" meaning "similar" or "like", and "-morphism", from Greek "morphe" meaning "form" or "shape", with the suffix "-ism" marking a process or relation. Put together, the term literally describes a relation between things of similar form. The meaning rests on that sense of likeness. In topology, two shapes are homeomorphic when one can be continuously stretched, bent, or squeezed into the other without tearing or gluing points together. A coffee mug and a doughnut are the classic pair: both have a single hole, so each can be deformed into the other. The mapping that accomplishes this is one-to-one and continuous in both directions, which is why the definition stresses that the properties it preserves are the essential ones. The term belongs to the technical vocabulary of topology, which developed in the late nineteenth and early twentieth centuries, so it is a modern coinage from classical roots rather than an ancient word.