Left and right uniformities

Reference no: EM13966713

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

Questions Cloud

Problem regarding the continuous real function : Show that there is a continuous real function f on X ,0 at x , and 1 on X \U . Use this function to show that x is not in the closure of K \U.

What distance does ball travel up and down on the bounce : What distance does the ball travel up and down on the bounce - How many bounces does it take for the ball to travel a total distance over 800 feet?

What happened to the current flowing from the source : Assume that the switch shown in the figure is closed, and calculate the current I, the power factor, and the real, reactive, and apparent power being supplied by the source.

How is social security currently funded : How is Social Security currently funded? What happens to funds once they are paid into the system? What changes have been made to the Social Security system from inception to date? Are those changes adequate?

Left and right uniformities : 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 an..

Topology of a separated uniformity : (a) Show that the uniform topology of a separated uniformity is always Hausdorff. (b) Show that if S has more than one point, a separated uniformity never converges as a filter for its product topology.

Base consisting of symmetric sets : 1. If X, Y , and Z are uniform spaces, f is uniformly continuous from X into Y , and g is uniformly continuous from Y into Z , prove that g ? f is uniformly continuous from X into Z. 2. Prove that any uniformity has a base consisting of symmetr..

What different types of reward do different characters seek : What different types of reward do the different characters seek? Please take into account how harsh conditions, treachery and death are woven into this myth.

Problem regarding the continuous real function : 1. A topological space (S, T ) is called perfectly normal iff for every closed set F there is a continuous real function f on S with F = f -1({0}).

User Account

All Pages