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Let $Y$ be a subset of $X$, with $X$ a metric space with metric $d$. Give an example where $A$ is open in $Y$, but not open in $X$.

Give an example where $A$ is closed in $Y$, but not closed in $X$.

For the first case, I can let $Y$ be the interval $[0,1]$ and $X$ be the interval $(0,1)$. How is this rigorously proved?

For the second case, I can let $Y$ be $(0,1)$ and $X$ be $[0,1]$.

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    Note if you take $Y = [0,1]$ and $X=(0,1)$, then you actually violate the condition that $y \subset X$.2011-11-06

1 Answers 1

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Try something like $Y=[0.5,0.75]$, $X=(0,1)$ and $A=Y$ for the example of a subset $A$ of $Y$ that is open in $Y$ but not in $X$.

For the second case try $Y=(0.5,0.75]$, $X=[0,1]$ and $A$ = $Y$.

I hope I understand your question correctly.