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: ${\ displaystyle X}$ ${\ displaystyle \ leq}$

 (R1) ${\ displaystyle \ forall x \ in X \ colon x \ leq x}$ Reflexivity (R2) ${\ displaystyle \ forall x, y, z \ in X \ colon (x \ leq y) \ land (y \ leq z) \ Rightarrow (x \ leq z)}$ Transitivity (R3) ${\ displaystyle \ forall x, y \ in X \ \ exists z \ in X \ colon (x \ leq z) \ land (y \ leq z)}$ 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 . ${\ displaystyle \ left (X, \ leq \ right)}$ ${\ displaystyle x \ leq y}$${\ displaystyle x}$${\ displaystyle y}$${\ displaystyle y}$${\ displaystyle x}$${\ displaystyle y \ geq x}$${\ displaystyle x \ leq y}$

Equivalently, one could also define a directed set as a quasi-order in which every finite subset has an upper bound.

Examples

• ${\ displaystyle X \ subseteq \ mathbb {R} ^ {n}; \, \ rho \ in \ mathbb {R} ^ {n} \ \ mathrm {fixed}; \, \ forall x, y \ in X :( x \ leq y): \ Leftrightarrow \ left \ | x- \ rho \ right \ | \ geq \ left \ | y- \ rho \ right \ | \ quad}$
(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 .${\ displaystyle {\ mathit {X}}}$${\ displaystyle {\ mathit {\ rho}}}$${\ displaystyle {\ mathit {\ rho}}}$${\ displaystyle {\ mathit {X}}}$ ${\ displaystyle f \ colon X \ to \ mathbb {R} ^ {n}}$${\ displaystyle x \ to \ rho}$
• ${\ displaystyle X = \ mathbb {N}; \, \ forall n, m \ in X: (n \ leq m): \ Leftrightarrow n \ mid m}$
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 .${\ displaystyle n}$ ${\ displaystyle m}$${\ displaystyle (\ mathbb {N}, \ mid)}$
• ${\ displaystyle X = \ mathbb {N}}$ with the usual less than or equal to ratio
• ${\ displaystyle X = \ mathbb {R}}$ with the usual less than or equal to ratio
With the help of this directed set, limit values ​​of functions or sequences for or , similar to the first example, can be understood as (network) convergences of their associated networks.${\ displaystyle x \ to \ infty}$${\ displaystyle n \ to \ infty}$
• ${\ displaystyle X = \ mathbb {N} ^ {2}; \, \ forall (n, m), (p, q) \ in X: ((n, m) \ leq (p, q)): \ Leftrightarrow (n \ leq p) \ land (m \ leq q)}$
With this direction on , the convergence of double sequences, again as network convergence, can be defined.${\ displaystyle \ mathbb {N} ^ {2}}$
• ${\ displaystyle M}$any set and (the power set )${\ displaystyle X = {\ mathcal {P}} (M)}$${\ displaystyle; \, \ forall A, B \ in X: (A \ leq B): \ Leftrightarrow A \ subseteq B}$
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

1. Heuser, pp. 249/250.