Canonical projections are open

Let $𝜋 𝛼 : 𝑋 ↠ 𝑋 𝛼$ be the canonical projections of the product topology. Then each $𝜋 𝛼$ is an open map. #m/thm/topology

Proof

Denote with $T 𝛼$ the topology of $𝑋 𝛼$ . Consider the subbasis $A 𝑋 = {𝜋 - 1 𝛼 𝑈 : 𝑈 \in T 𝛼 : 𝛼 \in 𝐴}$ of $𝑋$ , and let $𝑈 \in A 𝑋$ be a subbasic open set. Then $𝑈 = 𝜋 - 1 𝛼 (𝑉)$ for some $𝑉 \in T 𝛼$ , and since $𝜋 𝛼$ is surjective, $𝜋 𝛼 (𝜋 - 1 𝛼 (𝑉)) = 𝑉 = 𝜋 𝛼 (𝑈)$ . Hence $𝜋 𝛼 (𝑈)$ is open. Thus, by Proving open map with a subbasis, each $𝜋 𝛼$ is an open map.

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