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Let ϕ: [a,b] → D and ψ: [a,b] → D be piecewise smooth closed paths in a domain D . A homotopy from ϕ to ψ is a continuous function γ: [a,b]² to D satisfying
γ(a,t) = ϕ(t) ∀t ∈ [a,b] ,
γ(b,t) = ψ(t) ∀t ∈ [a,b] ,
∀s ∈ [a,b] , the path γ_s(t) = γ(s,t) is closed and piecewise smooth.
More generally, we may drop the closedness conditions.
A closed path is contractible if it is homotopic (i.e. there is a homotopy) to a constant path. A domain D is simply connected if every closed path is contractible.