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**HTale** I'm stuck on a question. It states to consider the set $\displaystyle Y = [-1,1]$ as a subspace of $\displaystyle \mathbb{R}$, it being the standard topology. And now I have to consider which are open in $\displaystyle \mathbb{R}$ and which are open in $\displaystyle Y$.

The first set is $\displaystyle A = \{x : 1/2 < |x| < 1\}$, which I think is $\displaystyle (1/2, 1) \cup (-1, -1/2)$. Now, that is open in $\displaystyle \mathbb{R}$, because it is the union of two open sets in $\displaystyle \mathbb{R}$. But is it open in $\displaystyle Y$? How do I tell if it is? How about the following cases:

$\displaystyle (1/2, 1] \cup [-1, -1/2)$

$\displaystyle [1/2, 1) \cup (-1, -1/2]$

$\displaystyle [1/2, 1] \cup [-1, -1/2]$

My guess is that they are all closed in $\displaystyle \mathbb{R}$, but all open in $\displaystyle Y$. Am I right? Why/why not?

Thanks in advance.