Mendelson's Introduction to Topology contains the following exercise (chapter 2, section 7, exercise 6:
Let $(Y,d^\prime)$ be a subspace of $(X,d)$. Prove that a subset $O^\prime\subset Y$ is an open subset of $(Y,d^\prime)$ if there is an open subset O of $(X,d)$ such that $O^\prime=Y\cap O$.
I tried to look at an example: X is the real line, Y is [0,2], $O=(1,3)$
then $O^\prime=(1,2]$ which does not appear to be open! So how is my example wrong?
The set $O'$ is not an open subset of $\mathbb R$, but it is an open subset of $[0,2]$, since, for each $x\in O'$, there is a $\varepsilon>0$ such that $(x-\varepsilon,x+\varepsilon)\cap[0,2]\subset O'$.