Let and be topological spaces with compact and let denote the projection of the topological product on ,.Given that is closed in , prove that is closed inX.

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- Feb 24th 2009, 08:18 AMAmanda1990Closedness in topological productLet and be topological spaces with compact and let denote the projection of the topological product on ,
*.*Given that is closed in , prove that is closed in*X*. - Feb 24th 2009, 06:58 PMaliceinwonderland
Tube lemma: Consider the product space , where Y is compact. If N is an open set containing the slice of , then N contains some tube about , where V is a neighborhood of in .

We shall show that is open in X using a tube lemma, which is equivalent to showing that is closed in X.

Let be any element in . By the tube lemma, there exists a tube such that

.

Now, for any in , we have an open set V containing such that . Thus, is open in .