Is a two-point set a retract of $\mathbb R^3$? I'm stuck with an example question. Can someone kindly give me some help?
Is a two point set a retract of $\mathbb R^3 $ ? 

The number of path components can only decrease under retractions.
Indeed, if $Y$ is a retract of $X$, then there is a surjective continuous map $X\to Y$, which induces a map from the set of path components of $X$ to that of $Y$ which is surjective.
