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Let A be a subset of a Euclidean space. Show that a subset U of A is open if for each point p in U, there exists an open set V in A containing p.

We have relative open sets. I know from point set topology that these corresponds to neighborhoods of points. If for such p's we intersect V with U and take their union we get the set U, but I dont understand how that makes U open in A.

nomeaning
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  • Since $U\subseteq A$, $V\cap U=V\cap A$ and $V\cap U$ is open in $A$. Does that resolve your difficulty? – saulspatz Apr 11 '21 at 12:52
  • I think it is because U is open in that union of neighborhoods and the union is also open in our original set A. Hence that makes U open in A. – nomeaning Apr 11 '21 at 21:39

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