Does the word "subspace" here imply "linear subspace"?
A subspace $Y$ of $\mathbb{R}$ is said to be a retract of $\mathbb{R}$ if there exists a continuous map $r: \mathbb{R} \to Y$ such that $r(y)= y$ for every $y \in Y$.
I am a bit confused by the definition. I need a bit naïve explanation.
A subspace in this context is a subset with the relative topology. For example, $[0,1]$ is a retract of $\mathbb R$ because $f(x)=x$ for $0 \leq x \leq 1$, $1$ for $x >1$ and $0$ for $x<0$ defines map with the stated properties.