I am studying The large scale structure of space-time by Hawking and Ellis. They use the sets $\partial A$ and $\dot{A}$; they seem to be both some kind of border of the set $A$, but they are different. Which is the difference between these two sets?
EDIT I thought it was a standard notation, so I did not included the definition:
$$\dot{A}=\bar{A}\cap\overline{(M-A)}$$
where $M$ is a manifold.
Why is this not the same as the boundary $\partial A$
The definition you added is the standard boundary of a set in a topological space. (In the context of general topology, $∂$ or $\operatorname{bd}$ is usually used). In the context of manifolds, $∂$ is used for the boundary of a manifold, which is different. For example if $A = M$, the topological boundary is always empty, but that is not the case for the manifold boundary. Also, if I'm not mistaken, the manifold boundary is the matter of the whole manifold rather than of a subset of the manifold.