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Pietro Majer
Pietro Majer
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A number of clever examples have been given of unbounded connected subsets of the Euclidean plane containing no infinite bounded subsets that are connected. None of those that I have seen are completely metrizable. Does anybody know if such can exist or if their existence can be ruled out by some theorem? I know that no such completely metrizable sets can exist if they are the graphs of functions of the form y=f(x) in the Cartesian plane. But does this prohibition extend to all unbounded connected planar sets?

A number of clever examples have been given of unbounded connected subsets of the Euclidean plane containing no bounded subsets that are connected. None of those that I have seen are completely metrizable. Does anybody know if such can exist or if their existence can be ruled out by some theorem? I know that no such completely metrizable sets can exist if they are the graphs of functions of the form y=f(x) in the Cartesian plane. But does this prohibition extend to all unbounded connected planar sets?

A number of clever examples have been given of unbounded connected subsets of the Euclidean plane containing no infinite bounded subsets that are connected. None of those that I have seen are completely metrizable. Does anybody know if such can exist or if their existence can be ruled out by some theorem? I know that no such completely metrizable sets can exist if they are the graphs of functions of the form y=f(x) in the Cartesian plane. But does this prohibition extend to all unbounded connected planar sets?

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Garabed Gulbenkian
Garabed Gulbenkian
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A question about unbounded connected subsets of the plane.

A number of clever examples have been given of unbounded connected subsets of the Euclidean plane containing no bounded subsets that are connected. None of those that I have seen are completely metrizable. Does anybody know if such can exist or if their existence can be ruled out by some theorem? I know that no such completely metrizable sets can exist if they are the graphs of functions of the form y=f(x) in the Cartesian plane. But does this prohibition extend to all unbounded connected planar sets?