Question #217053
Let X be a infinite set with cofinite topology .prove that any bijection from X to X is a homeomorphism
1
Expert's answer
2021-07-16T02:48:38-0400

Let XX be a infinite set with cofinite topology. Let us prove that any bijection f:XXf:X \to X is a homeomorphism. Let UXU\subset X be an open set. By definition of cofinite topology, we have that XUX\setminus U is finite. Since f:XXf: X\to X is a bijection, we conclude that f1(XU)=f1(X)f1(U)=Xf1(U)f^{-1}(X\setminus U)=f^{-1}(X)\setminus f^{-1}(U)=X\setminus f^{-1}(U) and f1(U)=Xf1(XU).f^{-1}( U)=X\setminus f^{-1}(X\setminus U). Since XUX\setminus U is finite, f1(XU)f^{-1}(X\setminus U) is also finite, and hence f1(U)f^{-1}( U) is cofinite. We conclude that f1(U)f^{-1}( U) belongs to the cofinite topology, that is f1(U)f^{-1}( U) is an open set. Therefore, ff is a continuous map. By analogy, for any open set VV its image f(V)f(V) is cofinite, and hence it is an open set in cofinite topology. We conclude that the map f:XXf:X\to X is open, and hence ff is a homeomorphism.



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