t implies u R t (by symmetry, it also implies t R u, as well as t R t and u R u) The logics that stem from these frame conditions are: K := no conditions D := serial T := reflexive B := reflexive and symmetric S4 := reflexive and transitive S5 := reflexive and Euclidean The Euclidean property along with reflexivity yields symmetry and transitivity. (The Euclidean property can be obtained, as well, from symmetry and transitivity.) Hence if the accessibility relation R is reflexive and Euclidean, R is provably symmetric and transitive as well. Hence for models of S5, R is an equivalence relation, because R is reflexive, symmetric and transitive. We can prove that these frames produce the same set of valid sentences as do the frames where all worlds can see all other worlds of W (i.e., where R is a "total" relation). This gives the corresponding modal graph which is total complete (i.e., no more edges (relations) can be added). For example, in any modal logic based on frame conditions: if and only if for some element u of G, it holds that and w R u. If we consider frames based on the total relation we can just say that if and only if for some element u of G, it holds that . We can drop the accessibility clause from the latter stipulation because in such total frames it is trivially true of all w and u that w R u. But note that this does not have to be the case in all S5 frames, which can still consist of multiple parts that are fully connected among themselves but still disconnected from each other. All of these logical systems can also be defined axiomatically, as is shown in the next section. For example, in S5, the axioms , and (corresponding to symmetry, transitivity and reflexivity, respectively) hold, whereas at least one of these axioms does not hold in each of the other, weaker logics. Topological semantics Modal logic has also been interpreted using topological structures. For instance, the Interior Semantics interprets formulas of modal logic as follows. A topological model is a tuple where is a topological space and is a valuation function which maps each atomic formula to some subset of . The basic interior semantics interprets formulas of modal logic as follows: iff iff iff and iff for some