G.Jäger [Compactness and connectedness as absolute properties in fuzzy topological spaces, Fuzzy sets and Systems 94(1998) 405-401] introduced a kind of (general) fuzzy topological space. In this paper, we propose a new kind of topological space in Šostak's sense, called Jäger-Šostak's I-fuzzy topological space, which reduced to Jäger's (general) fuzzy topological to two-valued logic. After that for each fuzzy subset of Jäger-Šostak's I-fuzzy topological space, we define a degree of connectedness, which overcome the deficit of study for the whole space a degree of being connected in public papers, and establish two characteristic theorems of the degree of being connectedness. Doing so we find that the degree of connectedness is an absolute property in Jäger- Šostak's I-fuzzy topology.