# Directed crowd

**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 .

## definition

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 .

- 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 .

## literature

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

## Individual evidence

- ↑ Heuser, pp. 249/250.