Exercise 24, § 4

Let G be a topological group, H a closed subgroup of G, $\pi : G \to G/H$ the canonical mapping. For every open symmetric neighbourhood V of e in G, denote by U_H(V) the set of pairs $(\dot{x}, \dot{y})$ of elements of G/H such that there exist $x \in \dot{x}$ and $y \in \dot{y}$ for which $x^{-1}y \in V$. The sets U_H(V) are invariant for the operation $(s, (\dot{x}, \dot{y})) \to (s.\dot{x}, s.\dot{y})$ of G in $(G/H) \times (G/H)$.

a) Now prove that if H is compact, the sets U_H(V) form a fundamental system of entourages for a uniform structure on G/H, compatible with the quotient topology (use exerc. 14 a) of III, p. 76).

b) Let G be the group $\mathbf{GL}(2, \mathbf{R})$ endowed with the topology defined in III, p. 73, exerc. 4, and let H be the closed commutative subgroup of G formed by the matrices $\begin{pmatrix} 1 & 0 \\ z & 1 \end{pmatrix}$, where z runs through $\mathbf{R}$. Now prove that the sets UH(V) do not form a fundamental system of entourages of a uniform structure on G/H (note that there are neighbourhoods V_0 of e in G such that, for no neighbourhood V of e in G, do we have VHV ⊂ HV_0H.)*