In a K-causal spacetime M, we know that the manifold topology can be recovered from the \(K^+\) order relation and therefore the chronological or causal future (past) of an event can be expressed in terms of K-causal order. In this article, we prove a conjecture put forward by R. D. Sorkin et al., which states that the chronological future \(I^+(p)\) of an event \(p\in M\) can be characterized in terms of K-causal order: \(K^+(p){\setminus } S=I^+(p)\) , where \(S=\lbrace r\in K^+(p)\mid\) every ’full chain’ from p to r meets \(\partial K^+(p)\rbrace\) and the same is true for \(I^-(p)\) ; that is, \(I^\pm (p)\) can be viewed as an order theoretic set instead of a set of events which are connected through a future (resp. past) directed time-like curves to p. In the same vein we show that if M is a continuous dcpo and \(\tilde{M}\) be its covering space, then the lift of a K-causal curve is also a K-causal curve.
