The future cone of a point is the set consisting of all points in the future of . (We usually interpret this condition weakly, so that itself belongs to its own future cone.)
Perhaps the terms forward cone resp. backward cone might be used as synonyms for the concept.
The dual concept is the past cone.
Given a Lorentzian manifold equipped with a global choice of which timelike curves are future-directed, the future of consists of all points such that there exists a future-directed curve (which may be timelike, lightlike, or a mixture) from to .
If we may put a global time coordinate on the manifold (which is a stronger condition), then the future cone of consists of all points such that and and are not space-like separated.
In the context of directed homotopy theory the future cone of a point, , in a directed topological space is, more-or-less, the directed subspace of those points which are greater than . In case the space might not be that ‘nice’ or, for instance, we have a local partial order rather than a global one, it is preferable to say that it is the directed space of points, , for which there is a distinguished path from to .