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I was tackling with this problem from Munkres: If Y is compact, then the projection map of $X \times Y$ is a closed map.

And I thought the same things as Akt904. After reading Brian M. Scott's comment I was nearly convinced but what about the open subset $\{(x,y): x^2+y^2<1\}$ of $\Bbb{R}\times\Bbb{R}$? Since it is open it is a union of basis elements, can it written as $A\times B$ where both $A$ and $B$ are open subsets of $\Bbb{R}$? Or it is not needed for open sets to be written as $A\times B$?

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up vote 2 down vote accepted

Open sets need not be products of open sets.

Note that if $U\subseteq X\times Y$ can be written as $A\times B$ then its projections onto $X$ and $Y$ are $A$ and $B$, respectively.

Note that if we project the open unit ball of $\Bbb R^2$ onto $\Bbb R$ we get $(-1,1)$ in both coordinates, their product is the open unit square, rather than the ball. Therefore it is not the product of any two subsets of $\Bbb R$, but rather the union of such sets.

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No, the set $\{(x,y): x^2+y^2<1\}$ can't be written as $A\times B$ where $A$ and $B$ are open subsets of $\mathbb{R}$; but yes, it is a union of such sets. There's no contradiction here.

Look at the Wikipedia page on the product topology.

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And the fact that the projection is open follows from all of these facts: function images preserve unions and all unions of open sets are open and projections of basic open sets are trivially open. – Henno Brandsma Jun 30 '13 at 9:33

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