Quotient maps in Banach spaces

Question

I came across two definitions of quotient maps in Banach spaces. A bounded linear transformation $T:X\to Y$ is a quotient map if:

A) $T(\textrm{int}(B_X))=\textrm{int}(B_Y)$

B) $\overline{T(B_X)}=B_Y$

I can see how a quotient map $T:X\to X/Z$ satisfies both $A)$ and $B)$, how $A)$ implies $T$ has norm $1$ and is surjective, and how $A)$ implies $B)$. I can also see how $B)$ implies that $T$ has norm $1$, but I cannot see how to prove that $T$ is surjective (or implies $A$, which would be the same thing). Any hints?

Answer 1

B is a sufficient condition to invoke the proof of the open mapping theorem (see Wiki). I guess that $T$ being open is sufficient to imply A.