The author presents a brief survey on known results from the theory of the continuous poset \(X\times {\mathbb R}\) of (generalized) formal balls of a metric space \((X,d)\). (Recall that the partial order \(\sqsubseteq\) is defined by \((x,r)\sqsubseteq (y,s)\) if \(d(x,y)\leq r-s.\))
In particular he considers results about the Lawson topology (the Martin topology, respectively) on \(X\times {\mathbb R}\) and compares these topologies with the product topology on \(X\times {\mathbb R},\) where \({\mathbb R}\) carries the Euclidean topology (resp. the Sorgenfrey topology). The concept of the hyperbolic topology of a metric space turns out to be useful in these discussions. Some open questions in the area are mentioned.