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

Let be a Hausdorff space and be a continuous injection. Then is Hausdorff. Similarly, if is a family of Hausdorff spaces, then the product and coproduct is also Hausdorff. #m/thm/topology

Proof

Then for any where , we have by injectivity and there exist disjoint open neighbourhoods of and respectively. Then are disjoint open neighbourhoods of respectively. Therefore is Hausdorff.

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