Dynamical systems over the space of upper semicontinuous fuzzy sets

Given a locally compact separable metric space E, a perfect transformation defined on E induces a Zadeh extension, which is a transformation from the space of all upper semicontinuous fuzzy sets defined on E to itself. Here, the latter space is equipped with the hit-or-miss topology. In this setting, each upper semicontinuous fuzzy set is identified with its hypograph, a closed subset in the product space of E and [0,1]. This approach does not require the relevant fuzzy sets to hold a compact support, and it also overcomes the drawback of the traditional level-set method. Further, it is proved that the Zadeh extension with this setting is continuous in the hit-or-miss topology; and dynamical properties of Zadeh extension regarding iteration have been explored.