Hit-and-far-miss topologies. (English) Zbl 1199.54069

Di Maio, Giuseppe; Naimpally, Somashekhar

Mat. Vesn. 60, No. 1, 59-78 (2008).

The paper is devoted to the investigation of so called hit-and-far-miss topologies (being a generalization of the Wijsman topology). The paper is split into four parts: preliminaries (explanations of the main hypertopologies in the context of metric spaces), hit-and-far-miss topologies (in the context of proximity and uniform spaces), new topologies (a long list of new hyperspace topologies constructed via fundamental “bricks” in metric and uniform spaces), graph topologies (hyperspace topologies on \(X\times Y\), restricted to the graphs of functions from \(X\) into \(Y\)). This unified approach makes some results seem more transparent and their proofs simpler.
Several suggestions for further investigation are formulated and new problems opened – among them: Is there a hypertopology which is not of hit-and-far-miss type?

Reviewer: Milosav Marjanović (Beograd)

MSC:

54B20	Hyperspaces in general topology
54E05	Proximity structures and generalizations
54E15	Uniform structures and generalizations

Keywords:

hypertopology; hit-and-far-miss topology; Hausdorff metric; Wijsman topology