Hausdorffness is preserved by subspaces, products, and coproducts, but not quotients

Let $𝑋$ be a Hausdorff space and $𝑖 : 𝑌 ↪ 𝑋$ be a continuous injection. Then $𝑌$ is Hausdorff. Similarly, if ${𝑋 𝛼} 𝛼 \in 𝐴$ is a family of Hausdorff spaces, then the product $\prod 𝛼 \in 𝐴 𝑋 𝛼$ and coproduct $∐ 𝛼 \in 𝐴 𝑋 𝛼$ is also Hausdorff. topology

Proof

Then for any $𝑥, 𝑦 \in 𝑌$ where $𝑥 \neq 𝑦$ , we have $𝑖 (𝑥) \neq 𝑖 (𝑦)$ by injectivity and there exist disjoint open neighbourhoods $𝑈, 𝑉 \subseteq 𝑋$ of $𝜄 (𝑥)$ and $𝜄 (𝑦)$ respectively. Then $𝑖 - 1 (𝑈), 𝑖 - 1 (𝑉)$ are disjoint open neighbourhoods of $𝑥, 𝑦$ respectively. Therefore $𝑌$ is Hausdorff.

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Hausdorffness is preserved by subspaces, products, and coproducts, but not quotients

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