Topological structures on graded sets

A group $L$ is called graded if it is represented as the union of a decreasing sequence of subgroups $L_m$. A general scheme for introducing the so-called sharp metric on such groups is proposed, with respect to which the algebraic operations are continuous and which is non-archimedean. It is shown that such a group is densely embedded in a complete group whose elements are series of a special type composed of elements of $L$. Similar constructions are considered for graded rings and graded vector spaces.
As examples, it is shown that in concrete special cases, the application of the described construction leads to the construction of $p$-adic numbers and to the construction of Taylor and Laurent series.