58       XII   TOPOLOGY AND TOPOLOGICAL ALGEBRA
PROBLEMS
1.	(a)   Let G be a topological group, K its identity component, H a subgroup of G
contained in K. Show that the connected components of the space G/H are the images
of the connected components of G under the canonical mapping tt : G -»-G/H. Show
that K is the smallest of the subgroups L of G such that G/L is totally disconnected.
(b)	Let G be the additive subgroup of the Banach space <^_R(N) (7.1.3) consisting of
the mappings n\-*f(ri) such that/(tf) e Q for all n e N and lim /(/?) exists in R. The
«-* 00
distance induced on G by that on ^_R(N) defines a topology compatible with the group
structure of G, and with respect to this topology G is totally disconnected. Let H be the
subgroup of G consisting of all /e G such that lim f(n) = 0. Show that H is closed
«~foo
in G and that G/H is isomorphic to R, and therefore connected.
(c)	Let G be a locally compact metrizable group, H a closed subgroup of G, and
tt : G ~> G/H the canonical mapping. Show that the connected components of G/H
are the closures of the images under tt of the connected components of G. (Reduce
to the case where G is totally disconnected, and use Problem 3 of Section 12.9.)
 2.	In the additive group R, let H be the subgroup Z and let A be the subgroup 0Z, where
6 is an irrational number. Show that the canonical bijection A/(A n H) -> (A + H)/H
is not an isomorphism of topological groups.
 3.	Let/? be a prime number and let (G_n) be an infinite sequence of topological groups all
equal to the discrete group Z/p^2Z. Let H_n be the subgroup pZ/p^2Z of G_n. Let G be the
subgroup of the product group fj G_n consisting of all x = (x_n) such that x_n e H_n for all
n
but a finite set of values of n. Let S3 be the set of neighborhoods of 0 in the product
group fj H_n = H c G. Show that S3 is a fundamental system of neighborhoods of 0
n
for a topology on G compatible with the group structure, and that in this topology G
is metrizable and locally compact and G/H is discrete. Show that the homomorphism
u: x\-+px of G into G is not a strict rnorphism of G into G, and that u(G) is not closed
inG.
 4.	Let G be a metrizable group, K a closed normal subgroup of G. If K and G/K are
complete, show that G is complete.
 5.	(a)   Let G be a topological group, K a normal subgroup of G. Show that if K and
G/K have no small subgroups (Section 12.9, Problem 6), then G has no small subgroups,
(b)   Deduce from (a) that if hi, H_2 are two normal subgroups of a topological group
G, such that g/h! and G/H_2 have no small subgroups, then G/(Hi n H_2) has no small
subgroups.
 6.	Let G be a connected, locally compact, metrizable group, K a compact normal sub-
group of G, and N a closed normal subgroup of K, such that K/N has no small sub-
groups (Section 12.9, Problem 6). Show that N is a normal subgroup of G. (Observe
that the hypothesis on K/N implies the existence of a neighborhood U of e in G such
that xNx"^1 «= N for all x e U.)
 7.	Let G be a connected metrizable group and H a compact normal commutative sub-
group of G, with no small subgroups. Show that H is contained in the center of G.