Justin Tatch Moore’s construction of a hereditarily Lindelöf, nonseparable space in ZFC connects set-theoretic topology with the combinatorics of minimal walks on omega_1. In this talk, we investigate the relationship between the Delta-property of Moore’s L-space L, and the structure of its associated oscillation tree.

After introducing C-sequences, lower traces, and oscillation functions, we will describe the space L and the tree T(o). We will prove that if

T(o) is special, then L is not a Delta-space. We then discuss a modification of the underlying C-sequence constructed in the forcing extension using a Cohen real and examine the stationary-set and oscillation arguments involved in obtaining a nonspecial tree.