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In the mathematical field of topology, a homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that 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. The word ''homeomorphism'' comes from the Greek words ''ὅμοιος'' (''homoios'') = similar and ''μορφή'' (''morphē'') = shape, form.〔Gamelin, T. W., & Greene, R. E. (1999). Introduction to topology. Courier Corporation. ()〕 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 torus are not. An often-repeated mathematical joke is that topologists can't tell the difference between a coffee cup and a donut, since a sufficiently pliable donut could be reshaped to the form of a coffee cup by creating a dimple and progressively enlarging it, while preserving the donut hole in a cup's handle. Topology is the study of those properties of objects that do not change when homeomorphisms are applied. ==Definition== A function ''f'': ''X'' → ''Y'' between two topological spaces (''X'', ''TX'') and (''Y'', ''TY'') is called a homeomorphism if it has the following properties: * ''f'' is a bijection (one-to-one and onto), * ''f'' is continuous, * the inverse function ''f'' −1 is continuous (f is an open mapping). A function with these three properties is sometimes called bicontinuous. If such a function exists, we say ''X'' and ''Y'' are homeomorphic. A self-homeomorphism is a homeomorphism of a topological space and itself. The homeomorphisms form an equivalence relation on the class of all topological spaces. The resulting equivalence classes are called homeomorphism classes. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「homeomorphism」の詳細全文を読む スポンサード リンク
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