HOMEOMORPHISM

in mathematics, a correspondence between two topological spaces by which topological properties are defined. A homeomorphism can be defined as a one-to-one correspondence between the points of two spaces such that corresponding sets have corresponding points of accumulation. Intuitively, two spaces are homeomorphic if one can be deformed into the other without tearing or folding. Two spaces are called topologically equivalent if there exists a homeomorphism between them. Topological properties are properties of a topological space that are also possessed by all other spaces homeomorphic to it. The properties of size and straightness in Euclidean space are not topological properties, while the connectedness of a figure is. Any polygon is homeomorphic to a circle, and these topologically equivalent figures are called simple closed curves. These curves have this topological property: they remain connected if one point is removed but become disconnected if two points are removed. Generally speaking, any property of a space that is defined in terms of relative position or the way in which points and sets are separated will be a topological property.