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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]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h1102001.png" /> and a collection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h1102002.png" /> of sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h1102003.png" />, one introduces a [[Topological structure (topology)|topological structure (topology)]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h1102004.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h1102005.png" /> depending on families <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h1102006.png" /> of sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h1102007.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h1102008.png" /> is closed under finite unions, by taking as an open [[Base|base]] for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h1102009.png" /> the family of sets of the form
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h11020010.png" /></td> </tr></table>
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h11020011.png" /></td> </tr></table>
+
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
  
The basic open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h11020012.png" /> collects those sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h11020013.png" /> that  "miss"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h11020014.png" /> and  "hit"  every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h11020015.png" />.
+
$$
 +
[ P  ^ {c} ;Q _ {1} \dots Q _ {k} ] =
 +
$$
 +
 
 +
$$
 +
=
 +
\left \{ {A \in {\mathcal A} } : {A \cap P = \emptyset,  A \cap Q _ {i} \neq \emptyset  ( i = 1 \dots k ) } \right \} .
 +
$$
 +
 
 +
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:
 
Important realizations of this scheme are:
  
1) the exponential topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h11020016.png" /> (cf. [[#References|[a1]]], [[#References|[a5]]], [[#References|[a9]]]);
+
1) the exponential topology $  \tau _ { {\mathcal F}, {\mathcal G} }  ( {\mathcal F} ) $(
 +
cf. [[#References|[a1]]], [[#References|[a5]]], [[#References|[a9]]]);
  
2) the hit-or-miss topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h11020017.png" /> (cf. [[#References|[a4]]]);
+
2) the hit-or-miss topology $  \tau _ { {\mathcal K}, {\mathcal G} }  ( {\mathcal F} ) $(
 +
cf. [[#References|[a4]]]);
  
3) the myope topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h11020018.png" /> (cf. [[#References|[a4]]]). Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h11020019.png" /> is the collection of all closed sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h11020020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h11020021.png" /> is the collection of all open sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h11020022.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h11020023.png" /> is the collection of all compact sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h11020024.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h11020025.png" /> (cf. [[Locally compact space|Locally compact space]]; [[Metric space|Metric space]]). The topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h11020026.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h11020027.png" /> converges in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h11020028.png" /> to a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h11020029.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h11020030.png" /> (meaning <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h11020031.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h11020032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h11020033.png" />; see [[#References|[a3]]]).
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h11020034.png" /> of the hit-or-miss topology to the collection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h11020035.png" />; the myope topology and the topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h11020036.png" /> coincide on any subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h11020037.png" /> that is compact in the myope topology (compactness of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h11020038.png" /> means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h11020039.png" /> is closed in the hit-or-miss topology and there exists a compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h11020040.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h11020041.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h11020042.png" />; cf. [[#References|[a4]]]).
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h11020043.png" /> on the collection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h11020044.png" /> is given by (cf. [[#References|[a2]]], [[#References|[a3]]]):
+
The Hausdorff metric $  D _ {H} $
 +
on the collection $  {\mathcal F} $
 +
is given by (cf. [[#References|[a2]]], [[#References|[a3]]]):
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h11020045.png" /></td> </tr></table>
+
$$
 +
D _ {H} ( A,B ) = \max  \left \{  \sup  _ {x \in A } { \mathop{\rm dist} } ( x,B ) ,  \sup  _ {y \in B } { \mathop{\rm dist} } ( y,A ) \right \} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h11020046.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h11020047.png" /> a bounded metric on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h11020048.png" />).
+
where $  { \mathop{\rm dist} } ( x,B ) = \inf  _ {y \in B }  d ( x,y ) $(
 +
$  d $
 +
a bounded metric on $  X $).
  
The topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h11020049.png" /> is metrizable (cf. [[Metrizable space|Metrizable space]]) by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h11020050.png" /> restricted to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h11020051.png" /> (cf. [[#References|[a3]]]); hence the hit-or-miss topology on any subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h11020052.png" /> that is compact in the myope topology is metrizable by the metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h11020053.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h11020054.png" /> from a metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h11020055.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h11020056.png" /> is upper semi-continuous (cf. also [[Semi-continuous mapping|Semi-continuous mapping]]) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h11020057.png" /> implies
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h11020058.png" /></td> </tr></table>
+
$$
 +
{\lim\limits  \sup } \Psi ( y _ {n} ) \subseteq \Psi ( y _ {0} ) .
 +
$$
  
An illustration is provided by the basic mappings of mathematical morphology in a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h11020059.png" /> (cf. [[#References|[a4]]], [[#References|[a7]]]): the opening <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h11020060.png" /> and the closing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110200/h11020061.png" /> (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]]]).
+
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]]]).
 
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]]]).

Revision as of 22:10, 5 June 2020


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=14982
This article was adapted from an original article by L. Polkowski (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article