Now, given a section f for the projection p
from this double-cover to Heath’s
V-space,
consider the set
X = {x∈R : f is
continuous in a neighborhood of
(x, 0)}.
For such x, by continuity, there are layer λ and
real ε > 0 such that
f carries Vε(x)
homeomorphically onto Vελ(x) (the full basic broken V of height ε
centered at f(x, 0) = (x, 0, λ)) .
Next, for each layer λ and ε > 0,
write
Xελ =
{x∈X :
f(Vε(x)) =
Vελ(x)}
.
Clearly
X =
∪{Xε0 :
ε > 0} ∪
∪{Xε1 :
ε > 0} =∪{X1/n0 :
n > 0} ∪
∪{X1/n1 :
n > 0} .
Alas, each Xελ is at most countable, as distinct reals x, y in
Xελ must lie at least 2ε
apart: |x - y| > 2ε. Indeed, ...
F.E.J. Linton • Wesleyan Univ. Math/CS Emeritus