Take a Mobius band $M$ and attached a disk $D^2$ along its boundary, which is a copy of $S^1$ embedded in $M$. It is known that what we obtain is something homeomorphic to the projective plane. Now the quotient $M/\partial M$ is also homeomorphic to the projective plane.
Are these two operations "equivalent"? More specifically, attaching a disk $D^2$ to a space $X$ and then contracting $D^2$ to a point is equivalent to taking the quotient of $X$ by $S^1$?