Order and topological structures of posets of the formal balls on metric spaces. (English) Zbl 1196.54048
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.
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.
Reviewer: Hans Peter Künzi (Rondebosch)
MSC:
54E35 | Metric spaces, metrizability |
46B20 | Geometry and structure of normed linear spaces |
54B10 | Product spaces in general topology |
54B20 | Hyperspaces in general topology |