For the Half-open line topology $(R,H)$. If $A = R - Q$, compute the closure, boundary, interior, and exterior for $A$.
I have $cl(A) = R$, $int(A) = \emptyset$, $bd(A) = R$, and $ext(A) = \emptyset$.
My logic each is below. Clearly, if I am wrong please let me kmow but I'm trying to ensure I'm understanding the topology correct and the required computations.
$R - Q$ is neither open nor closed in $(R,H)$, therefore, the smallest closed set containing $A$ is $R$ so $cl(A) = R$.
$int(A)$ is defined as the set of all points $x\in X$ for which there exists an open set $U$ such that $x \in U$ and $U \subseteq A$. $ext(A)$ is defined as the set of all points $x \in X$ for which there exists an open set $U$ such that $x \in U$ and $U \subseteq of X - A$.
$int(A)$ and $ext(A)$ are the empty set. All irrational numbers are surrounded by rational numbers so there exists no open sets $U$ that will not contain rational numbers. Therefore the open sets would not be a subset of $A$.
$bd(A)$ would be $R$ since $X = int(A) \cup bd(A) \cup ext(A)$ where $int(A)$ and $ext(A)$ are both empty sets. I know there is a more logical reason for this one but I cannot articulate it to myself.