  We consider a family of inverse limits of inverse sequences of closed unit
intervals with a single upper semi-continuous set-valued bonding function whose
graph is an arc; it is the union of two line segments in $[0,1]^2$, both of
them contain the origin $(0, 0)$, have positive slope, and extend to the
opposite boundary of $[0,1]^2$.
  We show that there is a large subfamily $\mathcal F$ of these bonding
functions such that for each $f\in \mathcal F$, the inverse limit of the
inverse sequence of closed unit intervals using $f$ as a single bonding
function, is homeomorphic to the Lelek fan.
