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It has been known for many years that if      X X       is a metrizable continuum then      2 X {2^X}       (the space of closed subsets of      X X      ) and      C ( X ) C(X)       (the subspace of connected members of      2 X {2^X}      ) are arcwise connected. These results are due to Borsuk and Mazurkiewicz [ l ] and J. L. Kelley [ 2 ], respectively. Quite recently M. M. McWaters [6] extended these theorems to the case of continua which are not necessarily metrizable, using Koch’s arc theorem for partially ordered spaces [ 3 ], [ 8 ]. In this note we prove these results for certain noncompact spaces by means of a simple generalization of Koch’s arc theorem.

Arcs in hyperspaces which are not compact