The infinite combinatorics developed in the previous chapters may be harnessed to give a treatment of regular variation in quite general contexts. Particularly useful tools here are the Category Embedding Theorem and the Effros Theorem. The main theorems of regular variation (see, e.g., BGT) include the Uniform Convergence Theorem (UCT) and the Characterization Theorem. The UCT is extended to the $L_1$-algebra of a locally compact metric group, using Reiter-like conditions from amenability. The Characterization Theorem can be formulated for normed groups X and H, with T a connected non-meagre Baire subgroup of the group of homeomorphisms from X to H. If for h : X → H is Baire and $h(tx)h(x)^{-1} \rightarrow k(t)$ for x → ∞ in X, then k is a continuous homomorphism from T to H. A calculus of regular variation is developed, involving the ‘differential modulus’. The theory is extended to the case of non-commutative H.