Left and right uniformities

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9. Let G be the affine group of the line, namely, the set of all transformations x f→ ax + b of R onto itself where a /= 0 and b ∈ R. Let G have its relative topology as {(a, b): a /= 0} in R2. Show that G is a topological group for which the left and right uniformities are different.

10. A metric d on an Abelian group G will be called an invariant metric iff d(x + z, y + z) = d(x, y) for all x, y, and z in G.

(a) Show that a metrizable Abelian topological group can be metrized by an invariant metric d.

(b) If an Abelian topological group G can be metrized by a metric for which it is complete, show that it is also complete for the invariant d from part (a). Hint: If not, take the completion of G for d and show it has an Abelian group structure extending that of G continuously; apply Theorem 2.5.4 and the category theorem (2.5.2).

Reference no: EM13966713

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