View source for Hit-or-miss topology

<!--
h1102001.png
$#A+1 = 61 n = 0
$#C+1 = 61 : ~/encyclopedia/old_files/data/H110/H.1100200 Hit\AAhor\AAhmiss topology
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

This scheme for introducing a topology into a collection of sets (cf. [[#References|[a1]]], [[#References|[a5]]], [[#References|[a9]]]) can be described conveniently by the  "hit or miss"  metaphor. Given a [[Topological space|topological space]]  $  ( X, \tau ) $
and a collection  $  {\mathcal A} $
of sets in  $  X $, 
one introduces a [[Topological structure (topology)|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|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. [[#References|[a1]]], [[#References|[a5]]], [[#References|[a9]]]);

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

3) the [[myope topology]]  $  \tau _ { {\mathcal F}, {\mathcal G} }  ( {\mathcal K} ) $(
cf. [[#References|[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|mathematical morphology]] (cf. [[#References|[a4]]], [[#References|[a7]]]) in Euclidean spaces, hence one most often considers locally compact metric spaces  $  X $(
cf. [[Locally compact space|Locally compact space]]; [[Metric space|Metric space]]). The topological space  $  ( {\mathcal F}, \tau _ { {\mathcal K}, {\mathcal G} }  ( {\mathcal F} ) ) $
is a compact metric space (cf. [[#References|[a4]]]); the topology of this space can be described (cf. [[#References|[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 [[#References|[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. [[#References|[a4]]]).

The Hausdorff metric  $  D _ {H} $
on the collection  $  {\mathcal F} $
is given by (cf. [[#References|[a2]]], [[#References|[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|Metrizable space]]) by  $  D _ {H} $
restricted to  $  {\mathcal K} \times {\mathcal K} $(
cf. [[#References|[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|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. [[#References|[a4]]], [[#References|[a7]]]): the opening  $  A _ {B} $
and the closing  $  A  ^ {B} $(
cf. also [[Mathematical morphology|Mathematical morphology]]). Both mappings are upper semi-continuous in the hit-or-miss topology (cf. [[#References|[a4]]]). The property of upper semi-continuity implies stability of either of these mappings in the morphological sense (cf. [[#References|[a7]]]).

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

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