3 – Coverings of a Topological Space
287
Cartesian product B × I equipped with the product topology induced by the
topology of B and the discrete topology of I (any subset of I is an open set);
conversely, the product X = B × I, where I is a discrete space, is a covering
space of B thanks to the map p(z, i) = z. Such a covering space is said to be
(globally) trivial or a decomposition.
In the case of algebraic functions (see end of the previous n
◦ ), there is a
“ n-sheeted ” covering space of B, but as shown by the graph x of the pseudofunction ζ = Log z studied in Chapter IV, § 4 in the general case, the number
of D i s is not necessarily finite or even constant if B is not connected.
10 Here
B = C
∗ , X is the set of couples (z, ζ) ∈ C
2 such that z = exp(ζ), and
p(z, ζ) = z. In any disc D ⊂ C
∗ , the “ multiform function ” Log z decomposes
into uniform branches L k (z) depending on k ∈ Z and the graphs D k ⊂
X of these L k are the connected components of p
−1 (D). As an aside, note
that this situation is just the same as that in example 1: associating the
point (e(ζ), ζ) of the graph of Log z to all ζ ∈ C, we get a homeomorphism
from the first covering space onto the second one which commutes with the
corresponding maps p. Two such covering spaces of a same space B are said
to be isomorphic..
In the general case, if condition (R) holds in a neighbourhood D of z,
it obviously also holds in any neighbourhood D
⊂ D. Hence if B is locally
connected, D can be assumed to be connected; then so are the D i . Since
the D i are pairwise disjoint, they are the connected components of the open
subset p
−1 (D) of X.
In practice, all the spaces considered are metrizable. If the topology of B
is defined by a distance d(x, y), then, for all z ∈ B, there exist numbers r > 0
such that X is trivial over the ball D(z, r). If r(z) denotes the upper bound of
these r, X is clearly trivial over D(z, r) for all r < r(z). If d(z, z
) < r < r(z),
X is trivial over D(z
, r
) for all r
< r(z) − r since D(z
, r
) ⊂ D(z, r). In
conclusion, r(z
) > r(z) − d(z, z
) and hence the function r is lower semicontinuous (Chap. V, n
◦ 10). For any compact subset K ⊂ B,
inf
z∈K
r(z) = r(K) > 0
(3.1)
because there exists z ∈ K, where r(z) is minimum (same reference).
(ii) Sections of a covering space. Since p : D i −→ D is a homeomorphism,
we can consider the inverse map ϕ i : D −→ D i ; it satisfies p ◦ ϕ i = id ; it
is the analogue of a local uniform branch. More generally, if E is a subset
of B, a section of X over E is any continuous map ϕ : E −→ X such that
p[ϕ(z)] = z for all z ∈ E, in other words, the analogue of a uniform branch on
E. Since p is a local homeomorphism, there are sections over all sufficiently
small neighbourhoods of all a ∈ B, namely the ϕ i ; there is even a section
10 Axiom (R) shows that the number of elements of p
−1 ({z}) is a locally constant
function of z ∈ B, and so is constant if B is connected. This number, whether
finite or not, is generally call the order of the covering space (X, B, p).
287
Cartesian product B × I equipped with the product topology induced by the
topology of B and the discrete topology of I (any subset of I is an open set);
conversely, the product X = B × I, where I is a discrete space, is a covering
space of B thanks to the map p(z, i) = z. Such a covering space is said to be
(globally) trivial or a decomposition.
In the case of algebraic functions (see end of the previous n
◦ ), there is a
“ n-sheeted ” covering space of B, but as shown by the graph x of the pseudofunction ζ = Log z studied in Chapter IV, § 4 in the general case, the number
of D i s is not necessarily finite or even constant if B is not connected.
10 Here
B = C
∗ , X is the set of couples (z, ζ) ∈ C
2 such that z = exp(ζ), and
p(z, ζ) = z. In any disc D ⊂ C
∗ , the “ multiform function ” Log z decomposes
into uniform branches L k (z) depending on k ∈ Z and the graphs D k ⊂
X of these L k are the connected components of p
−1 (D). As an aside, note
that this situation is just the same as that in example 1: associating the
point (e(ζ), ζ) of the graph of Log z to all ζ ∈ C, we get a homeomorphism
from the first covering space onto the second one which commutes with the
corresponding maps p. Two such covering spaces of a same space B are said
to be isomorphic..
In the general case, if condition (R) holds in a neighbourhood D of z,
it obviously also holds in any neighbourhood D
⊂ D. Hence if B is locally
connected, D can be assumed to be connected; then so are the D i . Since
the D i are pairwise disjoint, they are the connected components of the open
subset p
−1 (D) of X.
In practice, all the spaces considered are metrizable. If the topology of B
is defined by a distance d(x, y), then, for all z ∈ B, there exist numbers r > 0
such that X is trivial over the ball D(z, r). If r(z) denotes the upper bound of
these r, X is clearly trivial over D(z, r) for all r < r(z). If d(z, z
) < r < r(z),
X is trivial over D(z
, r
) for all r
< r(z) − r since D(z
, r
) ⊂ D(z, r). In
conclusion, r(z
) > r(z) − d(z, z
) and hence the function r is lower semicontinuous (Chap. V, n
◦ 10). For any compact subset K ⊂ B,
inf
z∈K
r(z) = r(K) > 0
(3.1)
because there exists z ∈ K, where r(z) is minimum (same reference).
(ii) Sections of a covering space. Since p : D i −→ D is a homeomorphism,
we can consider the inverse map ϕ i : D −→ D i ; it satisfies p ◦ ϕ i = id ; it
is the analogue of a local uniform branch. More generally, if E is a subset
of B, a section of X over E is any continuous map ϕ : E −→ X such that
p[ϕ(z)] = z for all z ∈ E, in other words, the analogue of a uniform branch on
E. Since p is a local homeomorphism, there are sections over all sufficiently
small neighbourhoods of all a ∈ B, namely the ϕ i ; there is even a section
10 Axiom (R) shows that the number of elements of p
−1 ({z}) is a locally constant
function of z ∈ B, and so is constant if B is connected. This number, whether
finite or not, is generally call the order of the covering space (X, B, p).