Two spaces are homeomorphic if there exists a continuous bijection with a continuous inverse. $\mathbb{R}$ is homeomorphic to any open interval $(a,b)$ — for example, $f: \mathbb{R}\to(0,4)$ via $f(x) = \dfrac{4}{1+e^{-x}}$ (a shifted logistic function) is a homeomorphism. $[-1,1]$ is compact while $\mathbb{R}$ is not, so they are not homeomorphic.