I am stuck with the following problem: X is a metric space. Suppose that Y is a subspace of X. Give an example that an open set in Y is not open in X.
My own approach was this: Suppose U is a subset of the rational numbers in the open set Y = (0,1). All elements in (0,1) is contained in real numbers. U is an open subspace of (0,1), but not in X.
Is my reasoning true? If not, could you please give me a clue? Thanks in advance!
Best Answer
What space are you using for $X$? I think this is an easier example:
Let $X = \Bbb{R}$ with the standard topology. Let $Y = (0,1]$. We know $\left( \frac{1}{2},\frac{3}{2} \right)$ is an open subset of $X$, so $Y \cap \left( \frac{1}{2},\frac{3}{2} \right) = \left( \frac{1}{2},1 \right]$ is open in $Y$. However, it is clear that $\left( \frac{1}{2},1 \right]$ is not open in $X$.