Question

Let f:(M,d)→(N,rho) be one-to-one and onto. Prove that f is a homeomorphism if and only if f(Aˉ)=f(A) for every subset A of M.

Answer 1

A function
`\( f: (M, d) \rightarrow (N, \rho) \)` is a homeomorphism if and only if
`\( f(\overline{A}) = f(A) \)` for every subset A of M.

Explanation:

A homeomorphism involves both continuity and bijectiveness. Firstly, if (f) is a homeomorphism, it implies that it's a continuous bijection, ensuring that the closure of A maps to the image of A. This is due to the preservation of continuity and bijection under a homeomorphism.

Conversely, if
`\( f(\overline{A})` = f(A) for all subsets A of M, it indicates that f is a bijection. This is because the equality of closures and images implies the surjectiveness of f, covering the entire codomain. Additionally, it guarantees the function's injectiveness as the pre-image of any element in the codomain has a unique pre-image in the domain.

This condition is crucial in ensuring the preservation of both open and closed sets. Homeomorphisms preserve the topological properties of spaces, and when
`\( f(\overline{A})` = f(A) holds, it implies that open sets are mapped to open sets and closed sets to closed sets, thereby maintaining the topological structure.

Furthermore, the equality
`\( f(\overline{A})` = f(A) ensures that f is not just a continuous map but also its inverse is continuous, fulfilling the definition of a homeomorphism where both f and
`f^(-1)` are continuous. Therefore, this equivalence between the closure of sets and their images is a fundamental criterion for characterizing homeomorphisms.