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Download A Set of Axioms for Differential Geometry by Veblen O., Whitehead J. H. PDF

By Veblen O., Whitehead J. H.

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Theorem. A Hausdorff group G has a completion if and only if every left Cauchy filter is a bilateral Cauchy filter. Proof. 9 the condition is sufficient, and by 4 . 5 it is necessary.. We shall normally denote the bilateral completion of a Hausdorff group G by G". 6, so we customarily identify HA with R . 15. 16 GA may be identified with lim,,(G/II). The following theorem applies, in particular, to the case where G1 = G". 12. Theorem. If G is a dense subgroup of a IIausdorff group G1, the closures in G1 of a fundamental system of neighborhoods of e in G form a fundamental system of neighborhoods o f e in G1.

Let G and G' be topological groups. (a) If E 2 G and if f: E->GI is both left and right uniformly continuous, then f is bilaterally uniformly continuous. (b) If a E G , the functions La: x->ax and Ra: x->xa are left, right, and bilaterally uniformly continuous from G to G. (c) The function j: x->x-l is bilaterally uniformly continuous from G to G. 2. Let f be the function from the additive topological group Q to the additive topological group R defined by f(x) = x2 Then f is continuous, the image under f of every Cauchy filter base on Q is a Cauchy filter base on R, and f = R, but f is not uniformly has a continuous extension t o continuous.

A A c ? m h & t L c on a set E is a function d from E X E to R such that for all x, y, z E E , d(x,x) = 0 , d(x,y) 2 0 , d(x,y) = d(y,x), and d(x,z) d(x,y) +d(y,z). Thus a semimetric d is a metric if and only if d(x,y) = 0 implies that x = y. If d is a semimetric on E , the inequality Id(x,y) -d(y,z) I I d(x,z) for all x, y , Z E E may be established just as for metrics. A semimetric d on E defines a topology just as a metric does. A fundamental system of neighborhoods of a E E is formed by all the balls B(a,r) of radius r > O about a, where B(a,r) = { X E E: d(a,x) C r].

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