This thesis considers metrics valued in abelian l -groups and their induced topologies. In addition to a metric into an l-group, one needs a filter in the positive cone to determine which balls are neighborhoods of their center. As a key special case, we discuss a topology on a lattice ordered abelian group from the metric and the filter of positives consisting of the weak units of G ; in the case of R , this is the Euclidean topology. Indeed, a major reason for considering these natural distance structures is that many l -groups topologies so arise, and we use this in later sections to analyze them. Next we show that every Tychonoff space can be derived from a generalized metric in the way described above, and that the converse holds. the following is the major result of this thesis. Theorem. (a) For each abelian l -groups G , Archimedean positive filter P on G , and G -metric d on a set is a Tychonoff topology. (b) Conversely, given a Tychonoff topological space , there is an abelian l -groups G, an Archimedean positive filter P on G , and a G -metric d on X such that .