On the retract of product of two spaces

Question

Let $X$ and $Y$ be two topological spaces. Let $R$ be a retract of $X\times Y$ with the retraction $r:X\times Y\longrightarrow R$. Assume that $\pi_1 :X\times Y\longrightarrow X$ and $\pi_2 :X\times Y\longrightarrow Y$ are projections maps.

Is $\pi_1 (R)$ a retract of $X$?

Answer 1

No, not in general. For instance, let $X=\mathbb{R}^2$ and $Y=\mathbb{R}$. Let $R\subset\mathbb{R}^3$ be helix-shaped path, so $R\cong [0,1]$ but $\pi_1(R)$ is a circle. Then $R$ is a retract of $\mathbb{R}^3$ (by the Tietze extension theorem, say) but the circle $\pi_1(R)$ is not a retract of $\mathbb{R}^2$.