A non-empty set is called directed if a relation (called direction ) is declared on it that satisfies the following requirements:
(R1)
Reflexivity
(R2)
Transitivity
(R3)
Existence of an upper bound
On a set it can make sense to define different directions (see examples ). In order to emphasize the intended direction, the ordered pair is also called a directed set . Speech for is before or after . Under one understands .
Equivalently, one could also define a directed set as a quasi-order in which every finite subset has an upper bound.
Examples
(Speaking: is directed towards, is directional center of .) One can use this direction to understand the limit value of a function for as (network) convergence of the associated network .
With the help of this directed set, limit values of functions or sequences for or , similar to the first example, can be understood as (network) convergences of their associated networks.
With this direction on , the convergence of double sequences, again as network convergence, can be defined.