Continuity concepts for correspondences

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Consider two metric spaces (X,d) and (Y,d') and a correspondence F from X to Y. Does a topology on the power set of Y, P(Y) exists such that F is upper (resp. lower hemi- continuous) if and only if F is continuous as a map from X to P(Y)?






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    Consider two metric spaces (X,d) and (Y,d') and a correspondence F from X to Y. Does a topology on the power set of Y, P(Y) exists such that F is upper (resp. lower hemi- continuous) if and only if F is continuous as a map from X to P(Y)?






    gn.general-topology continuity semicontinuity







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      Consider two metric spaces (X,d) and (Y,d') and a correspondence F from X to Y. Does a topology on the power set of Y, P(Y) exists such that F is upper (resp. lower hemi- continuous) if and only if F is continuous as a map from X to P(Y)?






      gn.general-topology continuity semicontinuity







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      Consider two metric spaces (X,d) and (Y,d') and a correspondence F from X to Y. Does a topology on the power set of Y, P(Y) exists such that F is upper (resp. lower hemi- continuous) if and only if F is continuous as a map from X to P(Y)?






      gn.general-topology continuity semicontinuity




      gn.general-topology continuity semicontinuity






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          Yes, these are the so-called Vietoris topologies. The upper Vietoris topology has a subbase consisting sets of the form $Fin 2^Ymid Ysubseteq O$ with $O$ open and the lower Vietoris topology has a subbase consisting sets of the form $Fin 2^Ymid Ycap Oneqemptyset$ with $O$ open. The correspondence is then upper hemicontinuous if and only if the corresponding function is continuous under the upper Vietoris topology, and the correspondence is lower hemicontinuous if and only if the corresponding function is continuous under the lower Vietoris topology. The right keyword for finding more on the subject is "hyperspace topologies."






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            Yes, these are the so-called Vietoris topologies. The upper Vietoris topology has a subbase consisting sets of the form $Fin 2^Ymid Ysubseteq O$ with $O$ open and the lower Vietoris topology has a subbase consisting sets of the form $Fin 2^Ymid Ycap Oneqemptyset$ with $O$ open. The correspondence is then upper hemicontinuous if and only if the corresponding function is continuous under the upper Vietoris topology, and the correspondence is lower hemicontinuous if and only if the corresponding function is continuous under the lower Vietoris topology. The right keyword for finding more on the subject is "hyperspace topologies."






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              Yes, these are the so-called Vietoris topologies. The upper Vietoris topology has a subbase consisting sets of the form $Fin 2^Ymid Ysubseteq O$ with $O$ open and the lower Vietoris topology has a subbase consisting sets of the form $Fin 2^Ymid Ycap Oneqemptyset$ with $O$ open. The correspondence is then upper hemicontinuous if and only if the corresponding function is continuous under the upper Vietoris topology, and the correspondence is lower hemicontinuous if and only if the corresponding function is continuous under the lower Vietoris topology. The right keyword for finding more on the subject is "hyperspace topologies."






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                Yes, these are the so-called Vietoris topologies. The upper Vietoris topology has a subbase consisting sets of the form $Fin 2^Ymid Ysubseteq O$ with $O$ open and the lower Vietoris topology has a subbase consisting sets of the form $Fin 2^Ymid Ycap Oneqemptyset$ with $O$ open. The correspondence is then upper hemicontinuous if and only if the corresponding function is continuous under the upper Vietoris topology, and the correspondence is lower hemicontinuous if and only if the corresponding function is continuous under the lower Vietoris topology. The right keyword for finding more on the subject is "hyperspace topologies."






                share|cite|improve this answer












                Yes, these are the so-called Vietoris topologies. The upper Vietoris topology has a subbase consisting sets of the form $Fin 2^Ymid Ysubseteq O$ with $O$ open and the lower Vietoris topology has a subbase consisting sets of the form $Fin 2^Ymid Ycap Oneqemptyset$ with $O$ open. The correspondence is then upper hemicontinuous if and only if the corresponding function is continuous under the upper Vietoris topology, and the correspondence is lower hemicontinuous if and only if the corresponding function is continuous under the lower Vietoris topology. The right keyword for finding more on the subject is "hyperspace topologies."







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                answered 2 hours ago









                Michael Greinecker

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