  The causal relation $K^+$ was introduced by Sorkin and Woolgar to extend the
standard causal analysis of $C^2$ spacetimes to those that are only $C^0$. Most
of their results also hold true in the case of spacetimes with degeneracies. In
this paper we seek to examine $K^+$ explicitly in the case of Lorentzian
topology changing Morse spacetimes containing isolated degeneracies. We first
demonstrate some interesting features of this relation in globally Lorentzian
spacetimes. In particular, we show that $K^+$ is robust and that it coincides
with the Seifert relation when the spacetime is stably causal. Moreover, the
Hawking and Sachs characterisation of causal continuity translates into a
natural expression in terms of $K^+$ for general spacetimes. We then examine
$K^+$ in topology changing Morse spacetimes both with and without the
degeneracies and find further characterisations of causal continuity.
