Topological property

Cardinality of a topology

Given a topological space the cardinality of the topology is a topological property, topology i.e. all the topologies of homeomorphic spaces have the same cardinality.

Proof

Let be a homeomorphism. Since The image map of a bijection is a bijection, is a bijection with inverse We can define since the preïmage of every open set must be open, which is clearly injective since it is restricted . Thus Using a similar argument, we can define an injection . Thus . Therefore .

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