closed range bounded linear operators

Question

Let $CL(X,Y)$ be the set of all closed range bounded linear operators from Banach space $X$ to Banach space $Y$. Is $CL(X,Y)$ an open set of $B(X,Y)$?

Answer 1

No, it isn't (assuming $B(X,Y) \ne CL(X,Y)$). For example, $0 \in CL(X,Y)$, but nonzero scalar multiples of any $T \in B(X,Y) \backslash CL(X,Y)$ are arbitrarily close to $0$.

Answer 2

On the positive side (a close true statement) : The set of all injective closed range bounded linear operators from the Banach space $X$ to the Banach space $Y$ is an open set of $B(X,Y)$. Indeed, these characterize as the operators $T\in B(X,Y)$ satisfying $\|Tx\|\ge C\|x\|$ for some constant $C=C(T)>0$, which property is clearly stable under small perturbations.