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Let $X$ be a subspace of a metric space $Y$. In general, if $A$ is open in $X$, then $A$ need not be open in $Y$. For example, in $\mathbb{R}^3$, an open disc on the $x$-$y$ plane is not open on $\mathbb{R}^3$.

But what if $X$ is an open subset of $Y$? Is $A$ necessarily open then? Intuitively I would say yes, but I can't prove it and can't find a counter example neither.

Spenser
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1 Answers1

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Yes, it’s true. Since $A$ is open in $X$, there is an open set $U$ in $Y$ such that $A=U\cap X$. If $X$ is also open in $Y$, then $U\cap X$ is the intersection of two open sets in $Y$ and is therefore also open in $Y$.

Brian M. Scott
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