In this paper we present a topological duality for a certain subclass of the $F_{\omega }$-structures defined by M. M. Fidel, which conform to a non-standard semantics for the paraconsistent N. C. A. da Costa logic $C_{\omega }$. Actually, the duality introduced here is focused on $F_{\omega }$-structures whose supports are chains. For our purposes, we characterize every $F_{\omega }$-chain by means of a new structure that we will call down-covered chain (DCC) here. This characterization will allow us to prove the dual equivalence between the...