The concept of a ∆-set of reals was originally defined by G.M. Reed. An equivalent version was defined by Eric van Douwe and later on was generalized to an arbitrary topology space, (∆-space) . Historically, this notion arose in the study of the normal Moore space conjecture, where Q-sets were used to construct important counterexamples to the conjecture. We prove that Moore’s L-space (a hereditarily Lindelöf but not separable space in ZFC) is not a Q-set space and if Aronszajn tree naturally associated with Moore’s L-space is special Moore’s L-space will not be a ∆-space.