A homeomorphism between two domains is a type of function that preserves the topological properties of the domains. Specifically, a homeomorphism is a continuous and bijective function that has a continuous inverse, meaning that it can be reversed without breaking continuity.
More formally, let X and Y be two topological spaces, and let f: X → Y be a function. Then f is a homeomorphism if the following conditions hold:
f is continuous, meaning that if x_n → x in X, then f(x_n) → f(x) in Y.
f is bijective, meaning that every element in Y has a unique preimage in X under f, and every element in X has a unique image in Y under f.
f has a continuous inverse function g: Y → X, meaning that if y_n → y in Y, then g(y_n) → g(y) in X.
If a homeomorphism exists between two domains, it means that they have the same topological properties, such as connectedness, compactness, and boundary structure. In other words, the domains are "topologically equivalent" and can be transformed into each other by a continuous, reversible function.
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