Fundamental group and the higher homotopy groups

They are basically defined as $\pi_n(X) = \{S^n \to X\} / \sim$

where $X$ is a Topological space, $S^n$ is the $n$-dimensional Hypersphere, the map $S^n \to X$ is continuous $/$ refers to the quotient operation, and $\sim$ refers to homotopic equivalence relation.