Integer topological charges for finite energy fields in the O(3) σ-model

In the (2+1)-dimensional classical O(3) σ-model, all finite energy fields have integer topological charges regardless of their asymptotic behavior at infinity. The topological charge is conserved for the fields with the finite Euclidean action, without further assumptions on the field equations or asymptotics. The fields with continuous first derivatives as well as fields in Sobolev-like spaces analogical to W^{1, 2} are considered.