Homeomorphism

A homeomorphism, also called a topological isomorphism or a bicontinuous function, is a bijective and continuous function between two topological spaces whose inverse is also continuous; the name was coined by the mathematician Henri Poincaré from Greek roots meaning similar shape. Homeomorphisms are the isomorphisms of the category of topological spaces, meaning they are exactly the mappings that preserve every topological property of the space they act on, and two spaces connected by a homeomorphism are called homeomorphic, meaning they are the same from a topological point of view. Roughly speaking, a homeomorphism results from continuously deforming one geometric object into another, so that a square and a circle are homeomorphic to each other while a sphere and a torus are not, though this informal picture can be misleading since some continuous deformations do not produce homeomorphisms and some homeomorphisms do not arise from continuous deformations at all.