Geodesic
Sequentially Relatively Uniformly Complete Riesz Spaces and Vulikh Algebras
Canadian journal of mathematics, Tome 24 (1972) no. 6, pp. 1110-1113

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Throughout this paper V will denote an Archimedean Riesz space with a weak unit e and a zero element θ. A sequence f 1,f 2,f 3, ... of points of V is said to converge relatively uniformly to a point f (with regulator the point g of V) if, for each ∈ > 0, there is a number N such that, if n is a positive integer and n > N, then |f — fn| < ∈g. In an Archimedean Riesz space a relatively uniformly convergent sequence has a unique limit. The sequence f 1, f 2, f 3, ... is called a relatively uniform Cauchy sequence (with regulator g) if, for each ∈ > 0, there is a number N such that if n and m are positive integers and n, m > N, then |fn — fm| < eg. A subset M of V is said to be sequentially relatively uniformly complete, written s.r.u.-complete, whenever every relatively uniform Cauchy sequence of points of M (with regulator in V) converges to a point of M. This property was defined by Luxemburg and Moore in [4] and some related conditions were derived. 
Tucker, C. T. Sequentially Relatively Uniformly Complete Riesz Spaces and Vulikh Algebras. Canadian journal of mathematics, Tome 24 (1972) no. 6, pp. 1110-1113. doi: 10.4153/CJM-1972-115-5
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