Let $A$ be the annulus in $\mathbb{C}$, what is the space $A/{\sim}$ generated by identifying all points on the "inner circle" with each other?

Question

The annulus is $A=\{z\in\mathbb{C}:1\leq |z| \leq 2\}$ and the "inner circle" here is the set of points $\{z\in\mathbb{C}:|z|=1\}$.

If we identify all points of the inner circle together, then, intuitively, the space is then homeomorphic to the closed disk. Is this correct?

If this is correct, how can I justify this answer beyond the observation that the quotient map has created some sort of "middle" for the annulus, thereby making it homeomorphic to the disk?