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R. Wijsman [[#References|[a4]]] introduced a convergence for sequences of proper lower semi-continuous convex functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130120/w1301201.png" />. Workers in topologies on hyperspaces found this convergence and the resulting topology quite useful and subsequently a vast body of literature developed on this topic (see [[#References|[a1]]], [[#References|[a2]]]).
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Suppose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130120/w1301202.png" /> is a [[Metric space|metric space]] and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130120/w1301203.png" /> denote the family of all non-empty closed subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130120/w1301204.png" />. For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130120/w1301205.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130120/w1301206.png" /> one sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130120/w1301207.png" />. One says that a net <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130120/w1301208.png" /> (cf. also [[Net (of sets in a topological space)|Net (of sets in a topological space)]]) is Wijsman convergent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130120/w1301209.png" /> if and only if for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130120/w13012010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130120/w13012011.png" />, i.e. the convergence is pointwise. The resulting topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130120/w13012012.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130120/w13012013.png" /> is called the Wijsman topology induced by the metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130120/w13012014.png" />. The dependence of the Wijsman topology on the metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130120/w13012015.png" /> is quite strong in as much as even two different uniformly equivalent metrics may induce different Wijsman topologies. Necessary and sufficient conditions for two metrics to induce the same Wijsman topology have been found by C. Costantini, S. Levi and J. Zieminska, among others. G. Beer showed that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130120/w13012016.png" /> is complete and separable (cf. also [[Complete metric space|Complete metric space]]; [[Separable space|Separable space]]), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130120/w13012017.png" /> is a Polish space, i.e. it is separable and has a compatible complete metric.
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If the pointwise convergence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130120/w13012018.png" /> is replaced by [[Uniform convergence|uniform convergence]], then Hausdorff convergence is obtained, which has been known for a long time. The associated Hausdorff topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130120/w13012019.png" /> is derived from the [[Hausdorff metric|Hausdorff metric]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130120/w13012020.png" /> given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130120/w13012021.png" />). It is known that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130120/w13012022.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130120/w13012023.png" /> is totally bounded (cf. also [[Totally-bounded space|Totally-bounded space]]).
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R. Wijsman [[#References|[a4]]] introduced a convergence for sequences of proper lower semi-continuous convex functions in ${\bf R} ^ { n }$. Workers in topologies on hyperspaces found this convergence and the resulting topology quite useful and subsequently a vast body of literature developed on this topic (see [[#References|[a1]]], [[#References|[a2]]]).
  
A natural question arises: What is the supremum of the Wijsman topologies induced by the family of all metrics that are topologically (respectively, uniformly) equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130120/w13012024.png" />. It was shown by Beer, Levi, A. Lechicki, and S. Naimpally that the supremum of topologically (respectively, uniformly) equivalent metrics is the Vietoris topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130120/w13012025.png" /> (cf. [[Exponential topology|Exponential topology]]; respectively, the proximal topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130120/w13012026.png" />). These are hit-and-miss type topologies; the former has been known for a long time while the latter is a rather recent discovery (1999; cf. also [[Hit-or-miss topology|Hit-or-miss topology]]). It is known that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130120/w13012027.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130120/w13012028.png" /> is compact, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130120/w13012029.png" /> is equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130120/w13012030.png" /> being totally bounded. G. Di Maio and Naimpally discovered a (hit-and-miss) proximal ball topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130120/w13012031.png" /> which equals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130120/w13012032.png" /> in almost convex metric spaces (these include normed linear spaces) [[#References|[a3]]]. L. Holá and R. Lucchetti have discovered necessary and sufficient conditions for the equality of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130120/w13012033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130120/w13012034.png" />.
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Suppose $( X , d )$ is a [[Metric space|metric space]] and let $\operatorname{CL} ( X )$ denote the family of all non-empty closed subsets of $X$. For each $x \in X$ and $A \in \operatorname{CL} ( X )$ one sets $d ( x , A ) = \operatorname { inf } \{ d ( x , a ) : a \in A \}$. One says that a net $A _ { \lambda } \in \operatorname{CL} ( X )$ (cf. also [[Net (of sets in a topological space)|Net (of sets in a topological space)]]) is Wijsman convergent to $A \in \operatorname{CL} ( X )$ if and only if for each $x \in X$, $d ( x , A _ { \lambda } ) \rightarrow d ( x , A )$, i.e. the convergence is pointwise. The resulting topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130120/w13012012.png"/> on $\operatorname{CL} ( X )$ is called the Wijsman topology induced by the metric $d$. The dependence of the Wijsman topology on the metric $d$ is quite strong in as much as even two different uniformly equivalent metrics may induce different Wijsman topologies. Necessary and sufficient conditions for two metrics to induce the same Wijsman topology have been found by C. Costantini, S. Levi and J. Zieminska, among others. G. Beer showed that if $( X , d )$ is complete and separable (cf. also [[Complete metric space|Complete metric space]]; [[Separable space|Separable space]]), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130120/w13012017.png"/> is a [[Polish space]], i.e. it is separable and has a compatible complete metric.
  
The Wijsman topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130120/w13012035.png" /> is always a Tikhonov topology (cf. also [[Tikhonov space|Tikhonov space]]) and a remarkable theorem of Levi and Lechicki shows that the separability of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130120/w13012036.png" /> is equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130120/w13012037.png" /> being metrizable or first countable or second countable.
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If the pointwise convergence $d ( x , A _ { \lambda } ) \rightarrow d ( x , A )$ is replaced by [[Uniform convergence|uniform convergence]], then Hausdorff convergence is obtained, which has been known for a long time. The associated Hausdorff topology $\mathcal{T} _ { \text{H}d }$ is derived from the [[Hausdorff metric|Hausdorff metric]] $d _ { \text{H} }$ given by $d _ { H } ( A , B ) = \operatorname { sup } \{ | d ( x , A ) - d ( x , B ) | : x \in X \}$). It is known that $T _ {  \operatorname{W} d } = T _ { \operatorname{H}d }$ if and only if $( X , d )$ is totally bounded (cf. also [[Totally-bounded space|Totally-bounded space]]).
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A natural question arises: What is the supremum of the Wijsman topologies induced by the family of all metrics that are topologically (respectively, uniformly) equivalent to $d$. It was shown by Beer, Levi, A. Lechicki, and S. Naimpally that the supremum of topologically (respectively, uniformly) equivalent metrics is the [[Vietoris topology]] $T _ { \text{V} }$ (cf. [[Exponential topology]]; respectively, the proximal topology $T _ { \delta }$). These are hit-and-miss type topologies; the former has been known for a long time while the latter is a rather recent discovery (1999; cf. also [[Hit-or-miss topology|Hit-or-miss topology]]). It is known that $T _ { W d } = T_{V}$ if and only if $( X , d )$ is compact, while $T _ { \text{W}d } = T _ { \delta }$ is equivalent to $( X , d )$ being totally bounded. G. Di Maio and Naimpally discovered a (hit-and-miss) proximal ball topology $T _ { \text{B} \delta }$ which equals $T_{\text{W}d}$ in almost convex metric spaces (these include normed linear spaces) [[#References|[a3]]]. L. Holá and R. Lucchetti have discovered necessary and sufficient conditions for the equality of $T_{\text{W}d}$ and $T _ { \text{B} \delta }$.
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The Wijsman topology $T_{\text{W}d}$ is always a Tikhonov topology (cf. also [[Tikhonov space|Tikhonov space]]) and a remarkable theorem of Levi and Lechicki shows that the separability of $X$ is equivalent to $T_{\text{W}d}$ being metrizable or first countable or second countable.
  
 
Wijsman's original work has been generalized by U. Mosco, Beer and others. Naimpally, Di Maio and Holá have studied Wijsman convergence in function spaces (see [[#References|[a2]]]).
 
Wijsman's original work has been generalized by U. Mosco, Beer and others. Naimpally, Di Maio and Holá have studied Wijsman convergence in function spaces (see [[#References|[a2]]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Beer,  "Topologies on closed and closed convex sets" , Kluwer Acad. Publ.  (1993)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Beer,  "Wijsman convergence: A survey"  ''Set-Valued Anal.'' , '''2'''  (1994)  pp. 77–94</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G. Di Maio,  S. Naimpally,  "Comparison of hypertopologies"  ''Rend. Ist. Mat. Univ. Trieste'' , '''22'''  (1990)  pp. 140–161</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R. Wijsman,  "Convergence ofsequences of convex sets, cones, and functions II"  ''Trans. Amer. Math. Soc.'' , '''123'''  (1966)  pp. 32–45</TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top">  G. Beer,  "Topologies on closed and closed convex sets" , Kluwer Acad. Publ.  (1993)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  G. Beer,  "Wijsman convergence: A survey"  ''Set-Valued Anal.'' , '''2'''  (1994)  pp. 77–94</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  G. Di Maio,  S. Naimpally,  "Comparison of hypertopologies"  ''Rend. Ist. Mat. Univ. Trieste'' , '''22'''  (1990)  pp. 140–161</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  R. Wijsman,  "Convergence ofsequences of convex sets, cones, and functions II"  ''Trans. Amer. Math. Soc.'' , '''123'''  (1966)  pp. 32–45</td></tr></table>

Latest revision as of 19:24, 1 January 2021

R. Wijsman [a4] introduced a convergence for sequences of proper lower semi-continuous convex functions in ${\bf R} ^ { n }$. Workers in topologies on hyperspaces found this convergence and the resulting topology quite useful and subsequently a vast body of literature developed on this topic (see [a1], [a2]).

Suppose $( X , d )$ is a metric space and let $\operatorname{CL} ( X )$ denote the family of all non-empty closed subsets of $X$. For each $x \in X$ and $A \in \operatorname{CL} ( X )$ one sets $d ( x , A ) = \operatorname { inf } \{ d ( x , a ) : a \in A \}$. One says that a net $A _ { \lambda } \in \operatorname{CL} ( X )$ (cf. also Net (of sets in a topological space)) is Wijsman convergent to $A \in \operatorname{CL} ( X )$ if and only if for each $x \in X$, $d ( x , A _ { \lambda } ) \rightarrow d ( x , A )$, i.e. the convergence is pointwise. The resulting topology on $\operatorname{CL} ( X )$ is called the Wijsman topology induced by the metric $d$. The dependence of the Wijsman topology on the metric $d$ is quite strong in as much as even two different uniformly equivalent metrics may induce different Wijsman topologies. Necessary and sufficient conditions for two metrics to induce the same Wijsman topology have been found by C. Costantini, S. Levi and J. Zieminska, among others. G. Beer showed that if $( X , d )$ is complete and separable (cf. also Complete metric space; Separable space), then is a Polish space, i.e. it is separable and has a compatible complete metric.

If the pointwise convergence $d ( x , A _ { \lambda } ) \rightarrow d ( x , A )$ is replaced by uniform convergence, then Hausdorff convergence is obtained, which has been known for a long time. The associated Hausdorff topology $\mathcal{T} _ { \text{H}d }$ is derived from the Hausdorff metric $d _ { \text{H} }$ given by $d _ { H } ( A , B ) = \operatorname { sup } \{ | d ( x , A ) - d ( x , B ) | : x \in X \}$). It is known that $T _ { \operatorname{W} d } = T _ { \operatorname{H}d }$ if and only if $( X , d )$ is totally bounded (cf. also Totally-bounded space).

A natural question arises: What is the supremum of the Wijsman topologies induced by the family of all metrics that are topologically (respectively, uniformly) equivalent to $d$. It was shown by Beer, Levi, A. Lechicki, and S. Naimpally that the supremum of topologically (respectively, uniformly) equivalent metrics is the Vietoris topology $T _ { \text{V} }$ (cf. Exponential topology; respectively, the proximal topology $T _ { \delta }$). These are hit-and-miss type topologies; the former has been known for a long time while the latter is a rather recent discovery (1999; cf. also Hit-or-miss topology). It is known that $T _ { W d } = T_{V}$ if and only if $( X , d )$ is compact, while $T _ { \text{W}d } = T _ { \delta }$ is equivalent to $( X , d )$ being totally bounded. G. Di Maio and Naimpally discovered a (hit-and-miss) proximal ball topology $T _ { \text{B} \delta }$ which equals $T_{\text{W}d}$ in almost convex metric spaces (these include normed linear spaces) [a3]. L. Holá and R. Lucchetti have discovered necessary and sufficient conditions for the equality of $T_{\text{W}d}$ and $T _ { \text{B} \delta }$.

The Wijsman topology $T_{\text{W}d}$ is always a Tikhonov topology (cf. also Tikhonov space) and a remarkable theorem of Levi and Lechicki shows that the separability of $X$ is equivalent to $T_{\text{W}d}$ being metrizable or first countable or second countable.

Wijsman's original work has been generalized by U. Mosco, Beer and others. Naimpally, Di Maio and Holá have studied Wijsman convergence in function spaces (see [a2]).

References

[a1] G. Beer, "Topologies on closed and closed convex sets" , Kluwer Acad. Publ. (1993)
[a2] G. Beer, "Wijsman convergence: A survey" Set-Valued Anal. , 2 (1994) pp. 77–94
[a3] G. Di Maio, S. Naimpally, "Comparison of hypertopologies" Rend. Ist. Mat. Univ. Trieste , 22 (1990) pp. 140–161
[a4] R. Wijsman, "Convergence ofsequences of convex sets, cones, and functions II" Trans. Amer. Math. Soc. , 123 (1966) pp. 32–45
How to Cite This Entry:
Wijsman convergence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wijsman_convergence&oldid=12771
This article was adapted from an original article by Som Naimpally (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article