Continuity concepts for correspondences

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











up vote
2
down vote

favorite












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)?










share|cite|improve this question







New contributor




Matteo Triossi is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.























    up vote
    2
    down vote

    favorite












    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)?










    share|cite|improve this question







    New contributor




    Matteo Triossi is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
    Check out our Code of Conduct.





















      up vote
      2
      down vote

      favorite









      up vote
      2
      down vote

      favorite











      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)?










      share|cite|improve this question







      New contributor




      Matteo Triossi is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.











      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






      share|cite|improve this question







      New contributor




      Matteo Triossi is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.











      share|cite|improve this question







      New contributor




      Matteo Triossi is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.









      share|cite|improve this question




      share|cite|improve this question






      New contributor




      Matteo Triossi is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.









      asked 3 hours ago









      Matteo Triossi

      111




      111




      New contributor




      Matteo Triossi is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.





      New contributor





      Matteo Triossi is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.






      Matteo Triossi is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.




















          1 Answer
          1






          active

          oldest

          votes

















          up vote
          2
          down vote













          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




















            Your Answer




            StackExchange.ifUsing("editor", function ()
            return StackExchange.using("mathjaxEditing", function ()
            StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
            StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
            );
            );
            , "mathjax-editing");

            StackExchange.ready(function()
            var channelOptions =
            tags: "".split(" "),
            id: "504"
            ;
            initTagRenderer("".split(" "), "".split(" "), channelOptions);

            StackExchange.using("externalEditor", function()
            // Have to fire editor after snippets, if snippets enabled
            if (StackExchange.settings.snippets.snippetsEnabled)
            StackExchange.using("snippets", function()
            createEditor();
            );

            else
            createEditor();

            );

            function createEditor()
            StackExchange.prepareEditor(
            heartbeatType: 'answer',
            convertImagesToLinks: true,
            noModals: false,
            showLowRepImageUploadWarning: true,
            reputationToPostImages: 10,
            bindNavPrevention: true,
            postfix: "",
            noCode: true, onDemand: true,
            discardSelector: ".discard-answer"
            ,immediatelyShowMarkdownHelp:true
            );



            );






            Matteo Triossi is a new contributor. Be nice, and check out our Code of Conduct.









             

            draft saved


            draft discarded


















            StackExchange.ready(
            function ()
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f312176%2fcontinuity-concepts-for-correspondences%23new-answer', 'question_page');

            );

            Post as a guest






























            1 Answer
            1






            active

            oldest

            votes








            1 Answer
            1






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes








            up vote
            2
            down vote













            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
























              up vote
              2
              down vote













              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






















                up vote
                2
                down vote










                up vote
                2
                down vote









                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."







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered 2 hours ago









                Michael Greinecker

                7,31923655




                7,31923655




















                    Matteo Triossi is a new contributor. Be nice, and check out our Code of Conduct.









                     

                    draft saved


                    draft discarded


















                    Matteo Triossi is a new contributor. Be nice, and check out our Code of Conduct.












                    Matteo Triossi is a new contributor. Be nice, and check out our Code of Conduct.











                    Matteo Triossi is a new contributor. Be nice, and check out our Code of Conduct.













                     


                    draft saved


                    draft discarded














                    StackExchange.ready(
                    function ()
                    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f312176%2fcontinuity-concepts-for-correspondences%23new-answer', 'question_page');

                    );

                    Post as a guest













































































                    Comments

                    Popular posts from this blog

                    What does second last employer means? [closed]

                    Installing NextGIS Connect into QGIS 3?

                    One-line joke