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 $ ( X, \tau ) $ and a collection $ {\mathcal A} $ of sets in $ X $, one introduces a topological structure (topology) $ \tau _ { {\mathcal P}, {\mathcal Q} } ( {\mathcal A} ) $ on $ {\mathcal A} $ depending on families $ {\mathcal P}, {\mathcal Q} $ of sets in $ X $, where $ {\mathcal P} $ is closed under finite unions, by taking as an open base for $ \tau _ { {\mathcal P}, {\mathcal Q} } ( {\mathcal A} ) $ the family of sets of the form

$$ [ P ^ {c} ;Q _ {1} \dots Q _ {k} ] = $$

$$ = \left \{ {A \in {\mathcal A} } : {A \cap P = \emptyset, A \cap Q _ {i} \neq \emptyset ( i = 1 \dots k ) } \right \} . $$

The basic open set $ [ P ^ {c} ;Q _ {1} \dots Q _ {k} ] $ collects those sets in $ {\mathcal A} $ that "miss" $ P $ and "hit" every $ Q _ {i} $.

Important realizations of this scheme are:

1) the exponential topology $ \tau _ { {\mathcal F}, {\mathcal G} } ( {\mathcal F} ) $( cf. [a1], [a5], [a9]);

2) the hit-or-miss topology $ \tau _ { {\mathcal K}, {\mathcal G} } ( {\mathcal F} ) $( cf. [a4]);

3) the myope topology $ \tau _ { {\mathcal F}, {\mathcal G} } ( {\mathcal K} ) $( cf. [a4]). Here, $ {\mathcal F} $ is the collection of all closed sets in $ X $, $ {\mathcal G} $ is the collection of all open sets in $ X $, and $ {\mathcal K} $ is the collection of all compact sets in $ X $.

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 $ X $( cf. Locally compact space; Metric space). The topological space $ ( {\mathcal F}, \tau _ { {\mathcal K}, {\mathcal G} } ( {\mathcal F} ) ) $ is a compact metric space (cf. [a4]); the topology of this space can be described (cf. [a4]) in terms of convergent sequences: A sequence $ ( F _ {n} ) _ {n} \subseteq {\mathcal F} $ converges in $ ( {\mathcal F}, \tau _ { {\mathcal K}, {\mathcal G} } ( {\mathcal F} ) ) $ to a set $ F \in {\mathcal F} $ if and only if $ F = {\lim\limits } F _ {n} $( meaning $ {\lim\limits \inf } F _ {n} = F = {\lim\limits \sup } F _ {n} $, where $ {\lim\limits \sup } F _ {n} = \cap _ {n} { {\cup _ {i} F _ {n + i } } bar } $ and $ {\lim\limits \inf } F _ {n} = \cap _ {( k _ {n} ) } {\lim\limits \sup } F _ {k _ {n} } $; 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 $ \tau _ { {\mathcal K}, {\mathcal G} } ( {\mathcal F} ) \mid _ {\mathcal K} $ of the hit-or-miss topology to the collection $ {\mathcal K} $; the myope topology and the topology $ \tau _ { {\mathcal K}, {\mathcal G} } ( {\mathcal F} ) \mid _ {\mathcal K} $ coincide on any subspace $ {\mathcal Z} \subseteq {\mathcal K} $ that is compact in the myope topology (compactness of $ {\mathcal Z} $ means that $ {\mathcal Z} $ is closed in the hit-or-miss topology and there exists a compact set $ K _ {0} $ such that $ K \subseteq K _ {0} $ for any $ K \in {\mathcal Z} $; cf. [a4]).

The Hausdorff metric $ D _ {H} $ on the collection $ {\mathcal F} $ is given by (cf. [a2], [a3]):

$$ D _ {H} ( A,B ) = \max \left \{ \sup _ {x \in A } { \mathop{\rm dist} } ( x,B ) , \sup _ {y \in B } { \mathop{\rm dist} } ( y,A ) \right \} , $$

where $ { \mathop{\rm dist} } ( x,B ) = \inf _ {y \in B } d ( x,y ) $( $ d $ a bounded metric on $ X $).

The topology $ \tau _ { {\mathcal F}, {\mathcal G} } ( {\mathcal K} ) $ is metrizable (cf. Metrizable space) by $ D _ {H} $ restricted to $ {\mathcal K} \times {\mathcal K} $( cf. [a3]); hence the hit-or-miss topology on any subspace $ {\mathcal Z} $ that is compact in the myope topology is metrizable by the metric $ D _ {H} $.

A mapping $ \Psi : {( Y,d ) } \rightarrow {( {\mathcal F} \tau _ { {\mathcal K}, {\mathcal G} } ( {\mathcal F} ) ) } $ from a metric space $ Y $ into $ {\mathcal F} $ is upper semi-continuous (cf. also Semi-continuous mapping) if $ y _ {n} \rightarrow y _ {0} $ implies

$$ {\lim\limits \sup } \Psi ( y _ {n} ) \subseteq \Psi ( y _ {0} ) . $$

An illustration is provided by the basic mappings of mathematical morphology in a Euclidean space $ ( {\mathcal E}, + ) $( cf. [a4], [a7]): the opening $ A _ {B} $ and the closing $ A ^ {B} $( 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