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.
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