For 𝑛 =0 the sphere fails to be path-connected as it consists of two disjoint points, hence for any 𝑝 ∈𝕊0 there exists only one loop with that basepoint, thus 𝜋1(𝕊0) ≅{𝑒}.

For 𝑛 =1 we regard a continuous loop 𝛼 as an endomorphism 𝛼 ∈𝖳𝗈𝗉(𝕊1,𝕊1). We claim that Degree of a circle endomorphism constitutes an isomorphism

This is well-defined and injective since Circle endomorphisms are homotopic iff they are of equal degree, and it is surjective because 𝛼 :𝑧 ↦𝑧𝑚 has degree 𝑚. Let 𝛼1,𝛼2 be paths with base 1 and let 𝜑1,𝜑2 :[0,1] →ℝ be the required continuous functions so that the following diagram commutes in 𝖳𝗈𝗉 for 𝑖 =1,2:

then the corresponding lift for the concatenated path 𝛼1 ∗𝑎2 is given by

and hence Φ[𝛼1][𝛽1] =𝜒(1) =𝜑1(1) +𝜑2(1) =Φ[𝛼1] +Φ[𝛼2]. Hence Φ is an isomorphism, so 𝜋1(𝕊1,1) =ℤ.

For 𝑛 ≥2 see Seifert-Van Kampen-Brown theorem.