PlanetMath: product topology preserves the Hausdorff property

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product topology preserves the Hausdorff property

(Theorem)

Theorem Suppose $\{X_\alpha\}_{\alpha\in A}$ is a collection of Hausdorff spaces. Then the generalized Cartesian product $\prod_{\alpha\in A} X_\alpha$ equipped with the product topology is a Hausdorff space.

Proof. Let $Y=\prod_{\alpha\in A} X_\alpha$ , and let be distinct points in . Then there is an index $\beta \in A$ such that $x(\beta)$ and $y(\beta)$ are distinct points in the Hausdorff space $X_\beta$ . It follows that there are open sets and in $X_\beta$ such that $x(\beta)\in U$ , $y(\beta) \in V$ , and $U\cap V = \emptyset$ . Let $\pi_\beta$ be the projection operator $Y\to X_\beta$ defined here. By the definition of the product topology, $\pi_\beta$ is continuous, so $\pi_\beta^{-1}(U)$ and $\pi_\beta^{-1}(V)$ are open sets in . Also, since the preimage commutes with set operations, we have that

$\displaystyle \pi_\beta^{-1}(U) \cap \pi_\beta^{-1}(V)$	$\displaystyle =$	$\displaystyle \pi_\beta^{-1} \big(U \cap V\big)$
	$\displaystyle =$	$\displaystyle \emptyset.$

Finally, since $x(\beta)\in U$ , i.e., $\pi_\beta(x)\in U$ , it follows that $x\in \pi_\beta^{-1}(U)$ . Similarly, $y\in \pi_\beta^{-1}(V)$ . We have shown that

and

are open disjoint neighborhoods of

respectively

. In other words,

is a Hausdorff space. $\Box$

"product topology preserves the Hausdorff property" is owned by archibal. [ owner history (1) ]

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Cross-references: neighborhoods, disjoint, open, continuous, operator, projection, open sets, index, points, proof, product topology, generalized Cartesian product, Hausdorff spaces, collection

This is version 4 of product topology preserves the Hausdorff property, born on 2003-05-31, modified 2004-03-12.
Object id is 4317, canonical name is ProductTopologyPreservesTheHausdorffProperty.
Accessed 2873 times total.

Classification:

AMS MSC:	54B10 (General topology :: Basic constructions :: Product spaces)
	54D10 (General topology :: Fairly general properties :: Lower separation axioms )

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