Directed crowd

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In mathematics, directed sets denote a generalization of non-empty, linearly ordered sets . They are used in topology to define networks and in category theory to define limits and colimites .


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.


(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 .
In the meaning of " shares ". The requirement (R3) is fulfilled by the smallest common multiple (lcm) . The directed amount is used for categorical limits , e.g. the per-finite numbers .
  • with the usual less than or equal to ratio
  • with the usual less than or equal to ratio
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.
  • any set and (the power set )
The requirement (R3) is fulfilled by the union .


Harro Heuser : Textbook of Analysis. Part 1. 15th edition. Teubner, Stuttgart a. a. 2003, ISBN 3-519-62233-5 .

Individual evidence

  1. Heuser, pp. 249/250.