The Lelek fan as the inverse limit of intervals with a single set-valued bonding function whose graph is an arc

Abstract

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.