The domain of influence [glo]domain of influence is define as the union of the past and future domain of influence. One of the main point in dealing with the causal structure is the notion of precede .
This relation is transitive, which means that if p precedes q and q precedes r then p precedes r.
Another property of this relation is linked with the past
and
future 
domain of influence
i.e.
the set of all points which precede p to the past
and the set of all points p precede to the future
(definitions
and page
). 
and 
are always open-sets.
Here we are more interested in causality violation . It is obvious that a closed timelike curve is an example of causality violation. A closed timelike curve through, say p, physically means that p precedes itself; therefore, p can influence itself!
The question, for our purpose of finding the topological structure of our spacetime, is to know if such anomalous spacetimes with causality violation can be ignored. It seems reasonable to say yes but another possibility is that our understanding of the phenomenon is not good enough at the moment, and later this causality violation will be found physically realizable. We don't know, and therefore, it can be interesting to see if there is some quantitative reason to rule out this causality violation.
One condition is that physical field can be made globally well defined. For example consider the Maxwell field. Is it possible to
make a global field from the local one? It turns out that it is
possible for the spacetimes with causality violation just in
very special cases. In most generic spacetimes which
violate causality, this property fails and then the causality
violation is physically unacceptable.
It might seem easy to determine whether there is causality violation in a spacetime: there are or not closed timelike curves. But it is not so simple. For example a spacetime can have no closed timelike curve but closed null curve which is also causally anomalous also. Even if we include null closed curves there is still a problem. Some spacetimes have almost closed timelike curves. This means that for some point p there is, in a small neighborhood, some timelike curve which begins and ends in this neighborhood. This leads to bad behavior that we also want to exclude.
All this motivates us to introduce the notion of stably
causal spacetimes(definition
page ), we open the light
cone and ask for no closed timelike curve in this opened
cone. Stably causal spacetimes have none of the above problems.
The stably causal condition has some important consequences like the fact that every stably causal spacetime is non-compact. In other word, the compact spacetimes are always subject to causality violation (it is the reason why they are not often considered in General Relativity). Another thing is that a spacetime admits a time-function if and only if it is stably causal. A time-function is a smooth scalar field t whose gradient is strictly timelike. It is often assign to be the time.
This last argument give us confidence that causally well-behaved spacetimes are the ones we need for Physics.
