  We demonstrate the breakdown of several fundamentals of Lorentzian causality
theory in low regularity. Most notably, chronological futures (defined
naturally using locally Lipschitz curves) may be non-open, and may differ from
the corresponding sets defined via piecewise $C^1$-curves. By refining the
notion of a causal bubble from [CG:12],we characterize spacetimes for which
such phenomena can occur, and also relate these to the possibility of deforming
causal curves of positive length into timelike curves (push-up). The phenomena
described here are, in particular, relevant for recent synthetic approaches to
low regularity Lorentzian geometry where, in the absence of a differentiable
structure, causality has to be based on locally Lipschitz curves.
