Let be a infinite set with cofinite topology. Let us prove that any bijection is a homeomorphism. Let be an open set. By definition of cofinite topology, we have that is finite. Since is a bijection, we conclude that and Since is finite, is also finite, and hence is cofinite. We conclude that belongs to the cofinite topology, that is is an open set. Therefore, is a continuous map. By analogy, for any open set its image is cofinite, and hence it is an open set in cofinite topology. We conclude that the map is open, and hence is a homeomorphism.
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