A partial set on \(X\) is an ordered couple \((A,B)\) of subsets of \(X\) such that \(A\cap B=\emptyset\). A DMF-algebra is a De Morgan algebra with a single fixed point for negation. Partial sets on \(X\) can be endowed with a structure of DMF-algebra, and any DMF-algebra is isomorphic to a field of partial sets. The concept of partial topological space on \(X\) is introduced taking as open sets partial sets, and a partial topological space is associated to every DMF-algebra \({\mathfrak A}\). It is proved that \({\mathfrak A}\) can be characterized as the set of compact clopens of the associated topological space.