Economics and game theory are replete with examples of parameterized games. We show that all minimal Nash payoff USCOs belonging to the Nash equilibrium correspondence of a parameterized game with payoff functions that are uniformly equicontinuous in players’ action choices with respect to parameters have minimal Nash USCOs that are essentially-valued as well as connected-valued. We also show that in general for any uniformly equicontinuous parameterized game, the Nash equilibrium correspondence is the composition of two correspondences: the graph correspondence of the collective security mapping and the Ky Fan Correspondence. The graph correspondence, a mapping from the parameter space into Ky Fan sets, encodes the specifics of the parameterized game being consider, while the Ky Fan Correspondence (i.e., the KFC), a mapping from Ky Fan sets into Nash equilibria, is universal and common to all parameterized games. We also show that the range of the graph correspondence, contained in the hyperspace of Ky Fan sets is a hyperspace Peano continuum - and is therefore locally connected. This means that for any two distinct Ky Fan sets contained in the range of graph correspondence there is a continuous segment in the range of the graph correspondence containing these two distinct Ky Fan sets as endpoints.