Regular completions of uniform convergence spaces

A regular completion with universal property is obtained for each member of the class of u–embedded uniform convergence spaces, a class which includes the Hausdorff uniform spaces. This completion is obtained by embedding each u-embedded uniform convergence space (X, I) into the dual space of a complete function algebra composed of the uniformly continuous functions from (X, I) into the real line.

Equicontinuity and -recursive transformation groups

In this paper we obtain results on equicontinuity and apply them to certain recursive properties of topological transformation groups (X, T, π) with uniform phase space X. For example, in the special case that each transition πt is uniformly continuous we consider the transformation group (X,Ψ,ρ), where Ψ is the closure of {πt|t∈T} in the space of all unimorphisms of X onto X with the topology of uniform convergence (space index topology) and p(x, φ) = φ(x) for (x, φ)∈ X × Ψ. (See (1), page 94, 11·18.) If π is a mapping on X × T we usually write ‘xt’ for ‘π(x, t)’ and ‘AT’ for ‘π(A × T)’ where A ⊂ X. In this case we obtain the following results:I. If (X, T, π) is almost periodic [regularly almost periodic] and πxis equicontinuous, then (X,Φ,ρ) is almost periodic [regularly almost periodic]II. Let A be a compact subset of X such that. If the left and right uniformities of T are equal and (X, T, π) is almost periodic [regularly almost periodic], then (X, T, π) is almost periodic [regularly almost periodic].