Some new properties

such as structural characterizations of open and closed sets

of the Stone space of all maximally consistent theories in the formal deductive system L

*

are investigated.A new three-valued topology on the set of all maximally consistent theories is introduced and it is proved that such three-valued topological space is zero-dimensional

covering-compact and Hausdorff.Lastly

the relationship between the three-valued topology and the Stone topology is clarified.All these results lay a foundation for the theory of reasoning about knowledge based on L

*

.