We can start our proof now.
Let be an opening covering of the product space.
Fix , for each , there exists an element in ,
denoted by which contains the point .
is an opening covering of , so by compactness
of , we may select a finite subcover, say .
Denote the which corresponds to by
and define
Now is an opening covering of , by assumption, there exists a countable subcover, denoted by
.
Finally, the family and is a countable open covering of the product space, and each element of this family is a subset of some element of . Q.E.D.
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