We proved a theorem in our functional analysis class showing that the subspace of bijective bounded linear operators between two Banach spaces $X$ and $Y$ is open in the space $B(X,Y)$ of bounded linear operators.
It was mentioned that the subspace of $(i)$ injective operators and $(ii)$ operators with dense range, however, are not open in $B(X,Y)$.
I tried coming up with counterexamples, but was not able to do so. Admittedly, I tried in finite dimension, and maybe the counterexamples come from infinite dimension? I'd like to see counterexamples to each of these cases. I'd appreciate some help in this, thanks.