For a double cover whose projection admits no continuous section,
we take two copies of the upper half-plane, topologized by ...

taking as basic open neighborhoods of points on either *x*-axis
the symmetric “broken open *V*s”, with fingers of equal length,
but with left finger in the same half-plane as the point on the *x*-axis,
and right finger in the other half-plane (omitting the corresponding point on
that other *x*-axis).

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