We can make a commutative diagram
Z2xZ2
Z3C x C
+a C+C
n n
[0, 1]2
3 [0, 2].
Recall that the left vertical map is given by
( ai2i,Eb`2')
2ai 2bi
33+t+i ' 3i+1)
and hence the diagonal composite is
ai2;, 1: bi2`) H 2E ai +bi
3i+t
Consequently, this composite has an image equal to the whole interval [0, 21.
Hence addition C x C --> [0, 2] is also surjective. A good way of viewing the situation is to make a picture of the subset C x C in the unit square of R2 and consider addition (x, y) H (x + y, 0) as a projection on the x-axis. The image of the totally disconnected set C x C is the whole interval [0, 2].
0
'. - .a .
0 1 2
A projection of C x C
0-
3.
Topological Algebra
3.1.
Topological Groups
Definition. A topological group is a group G equipped with a topology such that the map (x, y) H xy-t : G x G -* G is continuous.
r-.
.... i7' o0)
If G is a topological group, the inverse map x H x-1 is continuous (fix
x = e
in the continuous map (x, y) t-* xy-1) and hence a homeomorphism of order 2 of G. The translations x H ax (resp. x H xa) are also homeomorphisms (e.g., the inverse of x r- ax is x F+ a-tx). A subgroup of a topological group is a topological group for the induced topology.Examples. (1) With addition, ZP is a topological group. We have indeed
a'Ea+p"ZP, b'Eb+p"ZP==> a'-b'Ea-b+p"ZP
for all n > 0. In other words, using the p-adic metric (2.1). we have
Ix - a! S IP"I=p-", Iy - bI
<IP"I=P-"=I(x-y)-(a-b)I <p-",proving the continuity of the map (x, y) t--> x - y at any point (a, b).
(2) With respect to multiplication, ZP is a topological group. There is a funda- mental system of neighborhoods of its neutral element I consisting of subgroups:
1+pZpD 1+p2Zpj...D l+p"ZP3...
consists of subgroups: If a, 6 E Zp, we see that (1 + p"f)-t = 1 + p",8' for some ,6' E Zp (as in (1.5)), and hence
a=1+p"a, b=1+p",6 == ab-1=(1+p"a)(1+p",6')=I+P"y
for some y E ZP. Consequently,
a' E a(l + p"Zp), b' E b(1 + p"Zp) =
a'b'-t E ab-1(1 + p"Zp) (n > 1), and (x, y) H xy-1 is continuous. As seen in (1.5), 1 + pZp is a subgroup of index p - I in Zp Z. It is also open by definition (2.1). With respect to multiplication, all subgroups 1 + p"Zp (n > 1) are topological groups.(3) The real line R is an additive topological group.
If a topological group has one compact neighborhood of one point, then it is a locally compact space. If a topological group is metrizable, then it is a Hausdorff
space and has a countable fundamental system of neighborhoods of the neu- tral element. Conversely, one can show that these conditions are sufficient for metrizability.'
Let G be a metnzable topological group. Then there exists a metric d on G that defines the topology of G and is invariant under left translations:
d(gx, gy) = d(x, y).
'Specific references for the text are listed at the end of the book.
'C3
3. Topological Algebra 19
A metrizable group G can always be completed, namely, there exists a comp- lete group G and a homomorphism j : G --> G such that
the image j(G) is dense in G, j is a homeomorphism G -> j (G),
any continuous homomorphism f : G -> G' into a complete group G' can be uniquely factorized as f = g o j : G --p G - G' with a continuous homomor- phism g : G -> G'.
3.2.
Closed Subgroups of Topological Groups
As already observed, a subgroup of a topological group is automatically a topolo- gical group for the induced topology.
Lemma. Let G be a topological group, H a subgroup of G.
(a) The closure H of H is a subgroup of G.
(b) G is Hausdorff precisely when its neutral element is closed.
PROOF. (a) Let 1o : G x G G denote the continuous map (x, y) N xy-1. Since H is a subgroup, we have cp(H x H) C H and hence
rp(H x H)=cp(H x H)CSo(H x H)C H.
This proves that H is a subgroup.
(b) Let us recall that a topological space X is Hausdorff precisely when the diagonal AX is closed in the product space X x X. In any Hausdorff space the points are closed, and thus
G Hausdorff = (e} closed
AG = V-1(e) closed in G x G G Hausdorff.
The lemma is completely proved.
Proposition. Let H be a subgroup of a topological group G. If H contains a neighborhood of the neutral element in G, then H is both open and closed in G.
PROOF Let V be a neighborhood of the neutral element of G contained in H. Then for each h E H, h V is a neighborhood of h in G contained in H. This proves that H is a neighborhood of all of its elements, and hence is open in G. Consider now the cosets gH of H in G. Since translations are homeomorphisms of G, these cosets are open in G. Any union of such cosets is also open. But H is the complement of the union of all cosets gH ; H. Hence H is closed.
vii r.,(1. ,-T
Examples. The subgroups p"Zp (n > 0) are open and closed subgroups of the additive group Zp. The subgroups 1 + p"Zp (n > 1) are open and closed subgroups of the multiplicative group 1 + pZp.
Let us recall that a subspace Y of a topological space X is called locally closed (in X) when each point y E Y has an open neighborhood V in X such that Y fl v is closed in V. When this is so, the union of all such open neighborhoods of points of Y is an open set U in which Y is closed. This shows that the locally closed subsets of X are the intersections u fl F of an open set U and a closed set F of X. In fact, Y is locally closed in X precisely when Y is open in its closure Y.
Locally compact subsets of a Hausdorff space are locally closed (a compact subset is closed in a Hausdorff space). With this concept, the preceding proposition admits the following important generalization.
Theorem. Let G be a topological group and H a locally closed subgroup. Then H is closed.
PROOF If H is locally closed in G, then H is open in its closure H. But this closure is also a topological subgroup of G. Hence (by the preceding proposition) H is closed in H (hence H = H) and also closed in G by transitivity of this notion.
Alternatively, we could replace G by H, thus reducing the general case to H locally closed and dense in G. This case is particularly simple, since all cosets g H must meet H: g E H for all g E G, namely H = G.
Corollary 1. Let H be a locally compact subgroup of a Hausdorff topological group G. Then H is closed.
Corollary 2. Let r be a discrete subgroup of a Hausdorff topological group G.
Then r is closed.
The completion G of G is also a topological group. If G is locally compact, it must be closed in its completion, and we have obtained the following corollary.
Corollary 3. A locally compact metrizable group is complete.
3.3. Quotients of Topological Groups
As the following statement shows, the use of closed subgroups is well suited for constructing Hausdorff quotients. Let us recall that if H is a subgroup of a group G, then G /H is the set of cosets g H (g c G). The group G acts by left translations on this set. When H is a normal subgroup of G. this quotient is a group. Let now G be a topological group and
7r:G--> G/H
n`°
.G.v,' 3. Thpological Algebra 21denote the canonical projection. By definition of the quotient topology, the open sets U' C G/H are the subsets such that U = 7r-1 (U') is open in G. Now, if U is any open set in G, then
7r-1(7rU) = UH = U Uh
hEH
is open, and this proves that irU is open in G/H. Hence the canonical projection it : G -> G/H is a continuous and open map. By complementarity, we also see that the closed sets of G/H are the images of the closed sets of the form F = FH (i.e., F = 7ty1(F') for some complement F' of an open set U' C G/H). It is convenient to say that a subset A C G is saturated (with respect to the quotient map 7t) when A = A H, so that the closed sets of G/H are the images of the saturated closed sets of G (but 7r is not a closed map in general).
Proposition. Let H be a subgroup of a topological group G. Then the quotient G/H (equipped with the quotient topology) is Hausdorff precisely when H is closed.
PROOF Let it : G -> Gill denote the canonical projection (continuous by defi- nition of the quotient topology). If the quotient G/H is Hausdorff, then its points are closed and H = 7t-1(e) is also closed. Assume conversely that H is closed in G. The definition of the quotient topology shows that the canonical projection 7t is an open mapping. We infer that
7t2=7t x7t:GxG-G/H xG/H
is also an open map. But Ker(7r2) = H x H C G x G. Hence 7r2 induces a topological isomorphism
3f: (G x G)/(H x H) --> Gill x G/H.
To prove that G/H is Hausdorff, we have to prove that the diagonal
A={(x,x):xEG/H}
is closed in the Cartesian product Gill x G/H. Since the map n is a homeomor- phism, it is the same as proving that the inverse image A of this diagonal is closed in (G x G)/(H x H). This inverse image is
A={(g,k)modHxH:gH=kH}
={(g,k)modHxH:k`1gEH}.
But R = {(g, k) : k'1g E H} C G x G is closed by assumption: It is an inverse image of the closed set H under a continuous map. This closed set R is obviously saturated, i.e., satisfies
'17 X43
G1.
This proves that its image R' = A in the same quotient is closed, and the conclusion is attained.
Together with the theorem of the preceding section, this proposition establishes the following diagram of logical equivalences and implications for a topological group G and a subgroup H.
G/H finite Hausdorff H closed of finite index
u 4
Gill discrete e==> H open
4
4G/H Hausdorff H closed
3.4.
Closed Subgroups of the Additive Real Line
Let us review a few well-known results concerning the classical real line, viewed as an additive topological group. At first sight, the differences with Zp are striking, but a closer look will reveal formal similarities, for example when compact and discrete are interchanged.
Proposition 1. The discrete subgroups of R are the subgroups
aZ
(0 < a E R).PROOF. Let H {0} be a nontrivial discrete subgroup, hence closed by (3.2).
Consider any nonzero h in H, so that 0 < IhI (= ±h) E H. The intersection H fl [0, Ih I] is compact and discrete, hence finite, and there is a smallest positive element a E H. Obviously, Z - a C H. In fact, this inclusion is an equality. Indeed, if we
take any b E H and assume (without loss of generality) b > 0, we can write
b=ma+r (m EN, 0<r<a)
(take form the integral part of b/a). Since r = b - ma E H and 0 < r < a,
we must have r = 0 by construction. This proves b = ma E Z - a, and hence the reverse inclusion H C Z a.Corollary. The quotient of R by a nontrivial discrete subgroup H # (0} is compact.
Proposition 2. Any nondiscrete subgroup of R is dense.
PROOF. Let H C R be a nondiscrete subgroup. Then there exists a sequence of distinct elements h E H with h -* h E H. Hence s = 1h, - hl E H and,-, -* 0.
Since H is an additive subgroup, we must also have Z - 8 C H (for all n > 0), and the subgroup H is dense in R.
(f'C1+
w
0.O, Ilk 4.,
3. Topological Algebra 23
Corollary. (a) The only proper closed subgroups of R are the discrete sub- groups aZ (a E R).
(b) The only compact subgroup of R is the trivial subgroup (0}.
Using an isomorphism (of topological groups) between the additive real line and the positive multiplicative line, for example an exponential in base p
t i-+ p`, R -+ R>0
(the inverse isomorphism is the logarithm to the base p) we deduce parallel results for the closed (resp. discrete) subgroups of the topological group R>o.
Typically, we shall use the fact that the discrete nontrivial subgroups of this group have the form paZ (a > 0) or, putting 0 = p ", are the subgroups
OZ =(B' :In EZ}
for some 0 < 0 < 1.
3.5.
Closed Subgroups of the Additive Group of p-adic Integers
Proposition. The closed subgroups of the additive group Zp are ideals: They are
(0}, pmZp (m E N).
PROOF. We first observe that multiplication in ZP is separately continuous, since
Ix'a - xal = l all x' - xJ - 0 (x' - x).
Since an abelian group is a Z-module, if H C ZP is a closed subgroup, then for any h E H,
ZHCH ZpaCZaCH=H.
This proves thata closed subgroup is an ideal of Zp (or a Zp-module). Hence the result follows from (1.6).
Corollary 1. The quotient of Zp by a closed subgroup H # (0} is discrete.
Corollary 2. The only discrete subgroup of the additive group Zp is the trivial subgroup (0}.
PROOF. Indeed, discrete subgroups are closed: We have a complete list of these (being closedin ZP compact, a discrete subgroup is finite hence trivial). Alterna- tively, if a subgroup H contains a nonzero element h, it contains all multiples of h, and hence H D N h. In particular, HE) p"h -f 0 (n -+ oo). Since the elements
p"h are distinct, H is not discrete.
gyp. v,'
+
3.6.
Topological Rings
Definition. A topological ring A is a ring equipped with a topology such that the mappings
(x,y)Hx+y: AxA --> A, (x, y)Hx - y:AxA -fA
are continuous.
The second axiom implies in particular that y H -y is continuous (fix x = -l in the product). Combined with the first, it shows that
(x, y)Hx-y:AxA-*A
is continuous and the additive group of A is a topological group. A topological ring A is a ring with a topology such that A is an additive topological group and multiplication is continuous on A x A.
If A is a topological ring, the subgroup A" of units is not in general a to- pological group, since x H x` is not necessarily continuous for the induced topology (for an example of this, see the exercises). However, we can consider the embedding
xH(x,x-1):A"--* AxA,
and give A" the initial topology: It is finer than the topology induced by A. For this topology, A" is a topological group: The continuity of the inverse map, induced by the symmetry (x, y) H (y, x) of A x A, is now obvious. Still with this topology, the canonical embedding A" y A is continuous, but not a homeomorphism onto
its image in general.
Proposition. With the p-adic metric the ring ZP is a topological ring. It is a compact, complete, metrizable space.
PROOF. Since we already know that ZP is a topological group (3.1), it is enough to check the continuity of multiplication. Fix a and b in ZP and consider x = a + h, y = b + k in Z ,. Then
Ixy - abl = I(a+h)(b+k)-abl = Iak+hh - hkl
<max(IaI, Ibl)(Ihl + Ikl)+ Ihllki -j 0 (Ihl, Ikl -+ 0).
This proves the continuity of multiplication at any point (a, b) E ZP X Z,.
Corollary 1. The topological group ZP is a completion of the additive group Z equipped with the induced topology.
;T" T rte...A.. `^J
ago
A..
'v..fl via e"`
0p4
3. Topological Algebra 25 To make the completion process explicit, let us observe that if x = Y-i>o ai p`
is a p-adic number, then
X. = ai p` E N
O<i<n
defines a Cauchy sequence converging to x.
Corollary 2. The addition and multiplication of p-adic integers are the only continuous operations on ZP extending addition and multiplication of the nat- ural numbers.
3.7.
Topological Fields, Valued Fields
Definition. A topological field K is afield equipped with a topology such that the mappings
(x,y)Hx+y: KxK->K, (x,y)Hx.y: KxK -> K,
x i-> X-1 :
K" -> K"
are continuous.
Unless explicitly stated otherwise, fields are supposed to be commutative. A topological field is a topological ring for which K" = K - (0} with the induced topology is a topological group. Equivalently, a topological field is a field K equipped with a topology such that
(x, y) H x - y is continuous on K x K, (x, y) H x/y is continuous on K" x K".
Except for the appendix to Chapter II, we shall be interested only in valued fields:
Pairs (K, I. I) where K is a field, and 1. I an absolute value, namely a group homomorphism
I.I:K"--+R>o
extended by 101 = 0 and satisfying the triangle inequality Ix + yl 5 Ix! + jyl (x, Y E K), or the stronger ultrametric inequality
Ix+yl <max(IxI, IYI) (x, Y E K).
In this case d(x, y) = Ix - yI defines an invariant metric (or ultrametric) on K,
d(x, y) = d(x - a, y - a) = d(x - y, 0) (a, x, y E K).
.,(
as'
110,-1
This situation will be systematically considered from (11.1.3) on, and in the ap- pendix of Chapter II we shall show that any locally compact topological field can be considered canonically as a valued field.
Proposition 1. Let K be a valued field. For the topology defined by the metric d(x, y) = Ix - yI, K is a topological field.
PRooF. The map (x, y) H x - y is continuous. Let us check that the map (x, y) H xy-1 is continuous on K" x K'. We have
x + h x
_ by - kx
y+ k y y(y + k)
Hence if y 0 0 is fixed, Ikl < lyl/2, and c = max(IXI, Iyl),
x+h
xy+k
y < 2c Ihlly Ilkl -* 0 (Ihl, Ikl ---> 0).This proves that K is a topological field.
Proposition 2. Let K be a valued field. Then the completion k of K is again a valued field
PROOF. The completion K is obviously a topological ring, and inversion is contin- uous over the subset of invertible elements. We have to show that the completion
is afield. Let (xn) be a Cauchy sequence in K that defines a nonzero element of the completion K. This means that the sequence Ixn I does not converge to zero. There is a positive E > 0 together with an index N such that Ixn I > E for all n > N. The sequence (1/x, ),,>N is also a Cauchy sequence
I 1
Xn Xm
Xn - xm
XnXm < E-2I Xn - Xm I -* 0 (n, m -* no).
The sequence (1 /x,)n>N (completed with l's for n < N) defines an inverse of the original sequence (xx) in the completion K.