View source for Hit-or-miss topology

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

<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>

<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>

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" />.

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]]]);

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]]]);

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" />.

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 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 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]]]):

<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>

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" />).

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" />.

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

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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]]]).

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>