  Among the classical variants of the Pr\"ufer surface, some are homotopy
equivalent to a CW-complex (namely, a point or a wedge of a continuum of
circles) and some are not. The obstruction comes from the existence of
uncountably many `infinitesimal bridges' linking two metrizable open
subsurfaces inside the surface. We show that any non-metrizable surface that
possesses such a system of infinitesimal bridges cannot be homotopy equivalent
to a CW-complex. More than for the result on its own, we were motivated by
trying to blend elementary techniques of algebraic and set-theoretic topology.
