Alas, for any particular *i* and *n*, the set
*V*_{i, n}
is at most countable, because **distinct** points (*x*, 0)
and (*y*, 0) of one *V*_{i, n}
must lie at least 2/*n* apart, as measured by |*x* – *y*|.
Indeed, whatever the points (*x*, 0)
and (*y*, 0) of *V*_{i, n} ,
the one thing that is certain about the value |*x* – *y*| is
that, of the three classically exhaustive possibilities

| |*x* – *y*| | = 0 , |

0 < | |*x* – *y*| |
< 2/*n* , |

| |*x* – *y*| |
__>__ 2/*n* , |

only the first and third can subsist; for, were the second the case,
the section *f* must send the point at which the two intersecting fingers meet,
as the illustrations show, to a point in the double cover that lies at once
in **both** of the broken open *V*s shown,
which is impossible, as those two broken open *V*s are disjoint.

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