The Lelek Fan as the Inverse Limit of Intervals with a Single Set-Valued Bonding Function Whose Graph is an Arc

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; the graph is the union of two line segments in [ 0 , 1 ] 2 , both of which contain the origin (0, 0) and have positive slope. One of the segments extends to the top-boundary and the other to the right side boundary of [ 0 , 1 ] × [ 0 , 1 ] . We show that there is a large subfamily F of these bonding functions such that for each f ∈ 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.