Question 10

Question

What is a net? Given an example of an upward filtered ordered set.

Answer

A net is a function with a domain that has a directed set.

A net in $X$ is denoted by $x_{\bullet }=(x_i{)}_{i\in I}$
And is a function of the form $x_{\bullet }:I\to X$

What is the domain in a function?

The function $f:X\to Y$ would have the domain $X$. ($Y$ would be the codomain.)

Directed Set

A directed set is a non-empty set $I$ where the preorder (can be though as a direction) is usually assumed to be $\le$,which is upward directed.

So if you have $a,b\in A$ there exists some $c\in I$ such that

• $a\le c$
• $b\le c$