Answer to Question #217052 in Differential Geometry | Topology for Prathibha Rose

Question

Prove that the inverse of a.homeomorphism is also a.homeomorphism

Answer 1

$\text{By definition, a homeomorphism is a bijection such that both f and $f^{−1}$ are}\\\text{ continuous. As f is a bijection then by Bijection iff Inverse is Bijection, so is}\\\text{ $f^{−1}$. So by definition $f^{−1}$ is a bijection such that both $f^{−1}$ and $(f^{−1})^{−1}$ are}\\\text{ continuous. The result follows from Inverse of Inverse of Bijection}\\\text{which states that}\\\text{Let f:S→T be a bijection. Then: $(f^{−1})^{−1}=f$ where $f^{−1}$ is the inverse of f.}$