Editing Homeomorphism (section)

==Informal discussion==
The intuitive criterion of stretching, bending, cutting and gluing back together takes a certain amount of practice to apply correctly&mdash;it may not be obvious from the description above that deforming a [[line segment]] to a point is impermissible, for instance. It is thus important to realize that it is the formal definition given above that counts. In this case, for example, the line segment possesses infinitely many points, and therefore cannot be put into a bijection with a set containing only a finite number of points, including a single point.

This characterization of a homeomorphism often leads to a confusion with the concept of [[homotopy]], which is actually ''defined'' as a continuous deformation, but from one ''function'' to another, rather than one space to another. In the case of a homeomorphism, envisioning a continuous deformation is a mental tool for keeping track of which points on space ''X'' correspond to which points on ''Y''&mdash;one just follows them as ''X'' deforms. In the case of homotopy, the continuous deformation from one map to the other is of the essence, and it is also less restrictive, since none of the maps involved need to be one-to-one or onto. Homotopy does lead to a relation on spaces: [[homotopy equivalence]].

There is a name for the kind of deformation involved in visualizing a homeomorphism. It is (except when cutting and regluing are required) an [[homotopy|isotopy]] between the [[identity function|identity map]] on ''X'' and the homeomorphism from ''X'' to ''Y''.