H_n(X; R) = H_n(C_*(X; R)), where C_n(X; R) is the free R-module on continuous maps Δ^n → X. Functorial, homotopy-invariant, and agrees with simplicial/cellular/CW homology where defined.
H_n(X; R) = H_n(C_*(X; R)), where C_n(X; R) is the free R-module on continuous maps Δ^n → X. Functorial, homotopy-invariant, and agrees with simplicial/cellular/CW homology where defined.