Problem: Let $(X, \rho)$ be a metric space and $Y,E\subset X$.
If $Y$ is open, show that $cl_{Y}(E\ \cap Y)=cl_{X}(E)\ \cap Y$
And when I think about the problem, I am stuck in thinking "Is $(0,1)$ closed in $(0,1)$ or is $(0,0.5)$ closed in the $(0,1)$?" . I know that $(0,1)$ is closed in the $R$.
What is the closure in different sets?
I am sorry to ask this problem, but I am very confused.
This question doesn't need metric spaces. Since the question was tagged with general topology, I will assume that you know basic topology.
Hints:
(1) Notice that in a topological space $(X, \mathcal{T})$, $cl_\mathcal{T}(A)$ is the smallest closed set that contains the set $A$. That it is the smallest, implies that it is unique.
This follows from the fact that the closure is the intersection of all closed sets that contain $A$. You either have this available as a definition, or you should prove it.
(2) Show that $cl_X(E) \cap Y$ is the smallest closed set in the subspace $Y$ that contains $E\cap Y$. To show this, you will need the definition of subspace topology. Then use uniqueness.