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: Topological equivalence redirects here; see also [topological equivalence (dynamical systems)] .
A continuous deformation between a coffee mug and a doughnut illustrating that they are homeomorphic. But there does not need to be a continuous deformation for two spaces to be homeomorphic — only a continuous mapping with a continuous inverse.
In the [mathematical] field of [topology] , a homeomorphism or topological isomorphism or bicontinuous function (from the [Greek] words ( homoios ) = similar and ( morphē ) = shape, form) is a [continuous function] between two [topological space] s that has a continuous [inverse function] . Homeomorphisms are the [isomorphism] s in the [category of topological spaces] — that is, they are the [mappings] which preserve all the [topological properties] of a given space. Two spaces with a homeomorphism between them are called homeomorphic , and from a topological viewpoint they are the same.

Roughly speaking, a topological space is a [geometric] object, and the homeomorphism is a continuous stretching and bending of the object into a new shape. Thus, a [square] and a [circle] are homeomorphic to each other, but a [sphere] and a [doughnut] are not. An often-repeated joke is that topologists can't tell the coffee cup from which they are drinking from the doughnut they are eating, since a sufficiently pliable doughnut could be reshaped to the form of a coffee cup by creating a dimple and progressively enlarging it, while shrinking the hole into a handle.

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