Cardinality of a topology

Given a topological space $(𝑋, T)$ the cardinality of the topology $| T |$ is a topological property, #m/thm/topology i.e. all the topologies of homeomorphic spaces have the same cardinality.

Proof

Let $𝜙 : (𝑋, T 𝑋) \to (𝑌, T 𝑌)$ be a homeomorphism. Since The image map of a bijection is a bijection, $𝜙 ⋆ : P (𝑋) \to P (𝑌)$ is a bijection with inverse $𝜙 ⋆$ We can define $𝐹 : T 𝑌 ↠ T 𝑋 : 𝑉 \mapsto 𝜙 ⋆ (𝑉)$ since the preïmage of every open set $𝑉$ must be open, which is clearly injective since it is restricted $𝜙 ⋆$ . Thus $| T 𝑋 | \leq | T 𝑌 |$ Using a similar argument, we can define an injection $𝐺 : T 𝑋 ↠ T 𝑌 : 𝑉 \mapsto 𝜙 ⋆ (𝑉)$ . Thus $| T 𝑌 | \leq | T 𝑋 |$ . Therefore $| T 𝑋 | = | T 𝑌 |$ .

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