New York City Category Theory Seminar
Something Old, Something New, Something Borrowed, ...
Next F.E.J. Linton • Wesleyan Univ. Math/CS Emeritus

contradictory requirements on the section   Distinct reals x, y in Xελ must lie at least 2ε apart: |x − y| > 2ε. For if instead 0 < |x − y| < 2ε , then
Vε(x) ∩ Vε(y) = {(½(x+y), ½|x−y|)} ≠ ∅. 
while, up in the double cover,
∅ ≠ f(Vε(x) ∩ Vε(y)) ⊂ f(Vε(x)) ∩ f(Vε(y)) = Vελ(x) ∩ Vελ(y) = ∅, 
a contradiction. 
  With each Xελ thus at most countable, so is 
X = ∪{Xε0 : ε > 0} ∪ ∪{Xε1 : ε > 0} = ∪{X1/n0 : n > 0} ∪ ∪{X1/n1 : n > 0} .