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I got to know that the finite intersection of regular open sets in a topological space is regular. But what about "union of regular open set"? If this is regular I need the proof. And if this is not true, then I need a counter example.

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This is false. Take the intervals $(-1,0)$ and $(0,1)$. They are both regular open but the union is not since the closure is $[-1,1]$ whose interior is $(-1,1)\neq (-1,0) \cup (0,1)$.

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