Let $\displaystyle x_0 \in S^1$.

Is $\displaystyle S^1$ X $\displaystyle \{ x_0 \}$ a retract of $\displaystyle S^1$ X $\displaystyle S^1 $?

Is it a deformation retract of $\displaystyle S^1$ X $\displaystyle S^1 $?

I would think one would use the fact that if $\displaystyle S^1$ X $\displaystyle \{ x_0 \}$ were a deformation retract of $\displaystyle S^1$ X $\displaystyle S^1 $, then their fundamental groups would be isomorphic.

But the fundamental group of $\displaystyle S^1$ X $\displaystyle S^1 $ is ZXZ, while I think the fundamental group of $\displaystyle S^1$ X $\displaystyle \{ x_0 \}$ is ZX{0}. Would this be enough to justify it's not a deformation retract? If yes, I have a feeling that $\displaystyle S^1$ X $\displaystyle \{ x_0 \}$ is a retract of $\displaystyle S^1$ X $\displaystyle S^1 $. But how do you prove this?