Hit-or-miss topology

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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
How to Cite This Entry:
Hit-or-miss topology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hit-or-miss_topology&oldid=51364
This article was adapted from an original article by L. Polkowski (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article