Let $A\subseteq Y \subseteq X$ be metric spaces. What does Cl$_X(A)$ or Cl$_Y(A)$ mean?
For example, let $X=[0,10],Y=[1,9]$ and $A=(4,6)$.
Is $Cl_X(A)= Cl_Y(A)=[4,6]$?
I understand that $Int(X)=(0,10), Int(Y)=(1,9), Cl(A)=[4,6]$, but the notation is a little confusing when finding the closure of a subspace and I'm unsure what it means. This question was a little helpful for reference:
Best Answer
When $A$ is a subset of a topological space $X$ then the closure of $A$ is the smallest closed set of $X$ containing $A$. Analogously, the interior of $A$ is the largest open set of $X$ contained in $A$.
In cases where $A$ is a subset of more than one topological space of interest, as in the example $A\subseteq Y\subseteq X$ where $Y$ is a subspace of $X$, it is customary to note with respect to which topology the closure/interior is taken. i.e $Cl_X(A), Int_Y(A)$ etc.
This specification is important as even when $Y$ is a subspace of $X$ the closure of $A$ with respect to $Y$ may be different from the closure with respect to $X$. Same for interior. This happens because while the topologies of $Y$ and $X$ are intimately related, they are not the same topology, and hence have different closed and open subsets.
In your specific example, for instance, while $[1,9]$ is obviously not an open subset of $X$, it is an open subset of $Y$ (because it is the entire space). Therefore $Int_X(Y)$ is indeed $(1,9)$, but $Int_Y(Y)$ is $[1,9]$. Constructing an example where the closures differ is also possible, and not too difficult.