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Let $(X, \rho)$ be a compact metric space. Prove that $X \times X$ is compact in the product topology.

I am thinking that you need to construct an open cover of $X$, then obtain the finite subcover that follows from the compactness of $X$. Then somehow use the cross product of the open cover to make a finite subcover subcover of $X \times X$. I am just not sure exactly how notation etc should go.

Thank you!

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    I would simply call it product or Cartesian product, cross product is rather confusing and means something else. – 2017-02-02
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    http://math.stackexchange.com/questions/567335/cartesian-product-of-compact-sets-is-compact – 2017-02-02

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