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) .
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} .
Alas, each Xελ is at most countable, ...
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