Hit-or-miss topology

This scheme for introducing a topology into a collection of sets (cf. [a1], [a5], [a9]) can be described conveniently by the "hit or miss" metaphor. Given a topological space and a collection of sets in , one introduces a topological structure (topology) on depending on families of sets in , where is closed under finite unions, by taking as an open base for the family of sets of the form

The basic open set collects those sets in that "miss" and "hit" every .

Important realizations of this scheme are:

1) the exponential topology (cf. [a1], [a5], [a9]);

2) the hit-or-miss topology (cf. [a4]);

3) the myope topology (cf. [a4]). Here, is the collection of all closed sets in , is the collection of all open sets in , and is the collection of all compact sets in .

The hit-or-miss topology is an important tool in mathematical morphology (cf. [a4], [a7]) in Euclidean spaces, hence one most often considers locally compact metric spaces (cf. Locally compact space; Metric space). The topological space is a compact metric space (cf. [a4]); the topology of this space can be described (cf. [a4]) in terms of convergent sequences: A sequence converges in to a set if and only if (meaning , where and ; see [a3]).

The relations of the hit-or-miss topology 2) to the exponential topology 1) and myope topology 3) can be briefly summarized as follows. In general, the exponential topology is finer than the hit-or-miss topology 2) and the myope topology is finer than the restriction of the hit-or-miss topology to the collection ; the myope topology and the topology coincide on any subspace that is compact in the myope topology (compactness of means that is closed in the hit-or-miss topology and there exists a compact set such that for any ; cf. [a4]).

The Hausdorff metric on the collection is given by (cf. [a2], [a3]):

where ( a bounded metric on ).

The topology is metrizable (cf. Metrizable space) by restricted to (cf. [a3]); hence the hit-or-miss topology on any subspace that is compact in the myope topology is metrizable by the metric .

A mapping from a metric space into is upper semi-continuous (cf. also Semi-continuous mapping) if implies

An illustration is provided by the basic mappings of mathematical morphology in a Euclidean space (cf. [a4], [a7]): the opening and the closing (cf. also Mathematical morphology). Both mappings are upper semi-continuous in the hit-or-miss topology (cf. [a4]). The property of upper semi-continuity implies stability of either of these mappings in the morphological sense (cf. [a7]).

A ramification of the hit-or-miss topology was introduced into collections of rough sets generated from information systems (cf. [a6]) to yield a counterpart of mathematical morphology on abstract data sets (see also [a8]).

References

[a1]	G. Choquet, "Convergences" Ann. Univ. Grenoble , 23 (1948) pp. 55–112
[a2]	F. Hausdorff, "Grundzüge der Mengenlehre" , Leipzig (1914)
[a3]	K. Kuratowski, "Topology" , I–II , Acad. Press & PWN (1966–1968)
[a4]	G. Matheron, "Random sets and integral geometry" , Wiley (1975)
[a5]	E. Michael, "Topologies on spaces of subsets" Trans. Amer. Math. Soc. , 71 (1951) pp. 152–183
[a6]	L. Polkowski, "Mathematical morphology of rough sets" Bull. Polish Acad. Math. , 41 (1993) pp. 241–273
[a7]	J. Serra, "Image analysis and mathematical morphology" , Acad. Press (1982)
[a8]	A. Skowron, L. Polkowski, "Analytical morphology" Fundam. Inform. , 26–27 (1996) pp. 255–271
[a9]	L. Vietoris, "Stetige Mengen" Monatsh. Math. und Phys. , 31 (1921) pp. 173–204