If $A, B$ are open dense subsets of a metric space $X$, is their intersection dense?? We know normally intersection of two dense sets are not dense.
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Oops, did not see open. Yes. – orangeskid Feb 17 '15 at 15:49
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Note that it’s enough to know that one of the open sets is dense: see this question and its answers. – Brian M. Scott Feb 18 '15 at 01:06
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Let $U$ be any arbitrary open set. Then note that $U \bigcap (A \bigcap B) = (U \bigcap A )\bigcap B$. Here $U$ and $A$ are open sets so $(U \bigcap A )$ is open and $B$ being dense.....
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Can you please clarify how that is enough? Given $x \in X$, we need to show that $x$ belongs to closure of $A \bigcap B$... – Prajakta Bedekar Feb 17 '15 at 16:18