§ 4. Differential Manifolds
243
any continuous and injective map from a compact space X to a space Y is a
homeomorphism from X onto its image. To deduce that the map f : R −→
f (R) is open, it is, therefore, sufficient to check that so is the canonical map
from R onto R/Z, which is clear. Thus the image of f (R) is indeed a (compact)
submanifold of T
2 when a/b is rational.
In the general case, let us consider the homomorphism from R
2 onto T
2
given by
F (x, y) = (e(x), e(y))
(13.9)
and let D be the line of R
2 generated by the vector w = (a, b). As two points
of R
2 have the same image under F if and only if they differ by a point of Z
2 ,
the inverse image of Z = F (D) = f (R) under F is the subgroup G = D + Z
2 ,
i.e. the set of d+ω where d ∈ D, ω ∈ Z
2 . We show that when a/b is irrational,
G is everywhere dense in R
2 .
Let w
be a vector that is not in D ; the vector space R
2 is the direct sum
of D and the subspace D
generated by w
; hence
G = D + G ∩ D
,
so that it all amounts to showing that the subgroup G
= G ∩ D
of D
is
everywhere dense in D
. The set of t ∈ R such that tw
∈ G
is obviously a
subgroup H of the additive group R, and it all amounts to showing that it is
everywhere dense in R.
However, for such a subgroup, there are only four possibilities:
60
(a) H = {0},
(b) H = R,
(c) H is the set mZ of integer multiples of non-zero m ∈ R,
(d) H is everywhere dense in R.
Since cases (a) and (b) do not present any problems, first note that H contains
numbers t > 0 since t ∈ H implies −t ∈ H. Let m ≥ 0 be the infimum of
these numbers.
If m = 0, for r > 0, there exists t ∈ H such that 0 < t < r. However for
all x ∈ R and any real number t > 0, there exists q ∈ Z such that
|x − qt| < t .
Applying this remark to some t ∈ H such that 0 < t < r, this shows that we
are in case (d).
If m > 0, the same arguments show that, for all x ∈ H, there exists q ∈ Z
such that 0 ≤ x − qm < m, and so x − qm = 0 ; hence we are in case (c).
60 Other formulations : (i) every closed subgroup of R other than R is of the form
mZ ; (ii) every subgroup of R is either some Zm (possibly with m = 0), or is
everywhere dense in R.
243
any continuous and injective map from a compact space X to a space Y is a
homeomorphism from X onto its image. To deduce that the map f : R −→
f (R) is open, it is, therefore, sufficient to check that so is the canonical map
from R onto R/Z, which is clear. Thus the image of f (R) is indeed a (compact)
submanifold of T
2 when a/b is rational.
In the general case, let us consider the homomorphism from R
2 onto T
2
given by
F (x, y) = (e(x), e(y))
(13.9)
and let D be the line of R
2 generated by the vector w = (a, b). As two points
of R
2 have the same image under F if and only if they differ by a point of Z
2 ,
the inverse image of Z = F (D) = f (R) under F is the subgroup G = D + Z
2 ,
i.e. the set of d+ω where d ∈ D, ω ∈ Z
2 . We show that when a/b is irrational,
G is everywhere dense in R
2 .
Let w
be a vector that is not in D ; the vector space R
2 is the direct sum
of D and the subspace D
generated by w
; hence
G = D + G ∩ D
,
so that it all amounts to showing that the subgroup G
= G ∩ D
of D
is
everywhere dense in D
. The set of t ∈ R such that tw
∈ G
is obviously a
subgroup H of the additive group R, and it all amounts to showing that it is
everywhere dense in R.
However, for such a subgroup, there are only four possibilities:
60
(a) H = {0},
(b) H = R,
(c) H is the set mZ of integer multiples of non-zero m ∈ R,
(d) H is everywhere dense in R.
Since cases (a) and (b) do not present any problems, first note that H contains
numbers t > 0 since t ∈ H implies −t ∈ H. Let m ≥ 0 be the infimum of
these numbers.
If m = 0, for r > 0, there exists t ∈ H such that 0 < t < r. However for
all x ∈ R and any real number t > 0, there exists q ∈ Z such that
|x − qt| < t .
Applying this remark to some t ∈ H such that 0 < t < r, this shows that we
are in case (d).
If m > 0, the same arguments show that, for all x ∈ H, there exists q ∈ Z
such that 0 ≤ x − qm < m, and so x − qm = 0 ; hence we are in case (c).
60 Other formulations : (i) every closed subgroup of R other than R is of the form
mZ ; (ii) every subgroup of R is either some Zm (possibly with m = 0), or is
everywhere dense in R.