Answer to Question #217053 in Differential Geometry | Topology for Prathibha Rose

Question

Let X be a infinite set with cofinite topology .prove that any bijection from X to X is a homeomorphism

Answer 1

Let $X$ be a infinite set with cofinite topology. Let us prove that any bijection $f:X \to X$ is a homeomorphism. Let $U\subset X$ be an open set. By definition of cofinite topology, we have that $X\setminus U$ is finite. Since $f: X\to X$ is a bijection, we conclude that $f^{-1}(X\setminus U)=f^{-1}(X)\setminus f^{-1}(U)=X\setminus f^{-1}(U)$ and $f^{-1}( U)=X\setminus f^{-1}(X\setminus U).$ Since $X\setminus U$ is finite, $f^{-1}(X\setminus U)$ is also finite, and hence $f^{-1}( U)$ is cofinite. We conclude that $f^{-1}( U)$ belongs to the cofinite topology, that is $f^{-1}( U)$ is an open set. Therefore, $f$ is a continuous map. By analogy, for any open set $V$ its image $f(V)$ is cofinite, and hence it is an open set in cofinite topology. We conclude that the map $f:X\to X$ is open, and hence $f$ is a homeomorphism.