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Show that "the Cartesian product of infinite number of sets is uncountable" is not valid if all but finitely many of the sets have cardinality 1.

Could anyone clarify this statement for me please?

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  • $\begingroup$ What is 1 times 1 ? How many ordered pairs pairs are there whole first component is "blue" and whose second is "3" ? $\endgroup$ – kimchi lover Aug 14 '17 at 23:55
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Note that $\prod_{i\in[0,1]}\{i\}$ is an infinite (actually uncountable!) product of sets, but it only contains one element; namely the function $f:[0,1]\to[0,1]$ defined by $f(i)=i$.

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