Definition:Net (Preordered Set)
 | It has been suggested that this page or section be merged into Definition:Moore-Smith Sequence. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Mergeto}} from the code. |
 | This page has been identified as a candidate for refactoring of medium complexity. In particular: Definitions are not equivalent! Until this has been finished, please leave {{Refactor}} in the code.
New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only. Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Refactor}} from the code. |
 | This article needs to be linked to other articles. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{MissingLinks}} from the code. |
Definition
Let $X$ be a nonempty set.
Let $\struct {\Lambda, \precsim}$ be a preordered set.
Let $F: \Lambda \to X$ be a mapping.
Then $F$ is referred to as a net.
Other Definition
Let $X$ be a set.
Let $\struct {D, \le}$ be a directed set.
A mapping $\phi: D \to X$ from $D$ to $X$ is called a net in $X$.
It is common to write $\map \phi d = x_d$, and subsequently denote the net $\phi$ by $\family {x_d}_{d \mathop \in D}$, mimicking the notation for indexed sets and sequences.
The first definition is not equivalent to this one because a directed set is more than a preordered set.
For example, $\struct {\set {a, b}, \le}$, in which the relation is $a \le a$ and $b \le b$ is a preorder, but not a directed set.
Also see
Work in progress
 | This page or section has statements made on it that ought to be extracted and proved in a Theorem page. In particular: Extract the information in this note into a separate page expanding this point into a specific result. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by creating any appropriate Theorem pages that may be needed. To discuss this page in more detail, feel free to use the talk page. |
Note : Nets are extensions of sequences. In fact, a sequence over a set $X$ is a mapping from $\N$ to $X$ and $\N$ - endowed with the standard comparison relation $\le$ (which is a partial order and a fortiori a preorder) - is a preordered set. Hence a sequence is a special case of a net.
Sources
 | There are no source works cited for this page. Source citations are highly desirable, and mandatory for all definition pages. Definition pages whose content is wholly or partly unsourced are in danger of having such content deleted. To discuss this page in more detail, feel free to use the talk page. |