Sequences and Nets

# Sequences and Nets

Recall from the Directed Sets page that a directed set is a pair $(X, \leq)$ where $X$ is a nonempty set and $\leq$ is a reflexive transitive binary relation on $X$ with the additional property that every pair of elements in $X$ has an upper bound in $X$.

The concept of a direct set is important in formulating the theory of sequences and nets.

## Sequences

 Definition: Let $X$ be a nonempty set. A Sequence on $X$ is a function $S$ whose domain is the set of natural numbers and is commonly written as $S = (S_1, S_2, ...)$. The $n^{\mathrm{th}}$ Term of $S$ is the value $S(n)$, which is also denoted $S_n$.

## Nets

 Definition: Let $X$ be a nonempty set. A Net on $X$ is a pair $(S, \leq)$ where $S$ is a function and $\leq$ is a binary relation which directs $\mathrm{dom}(S)$.

Here, the notation "$\mathrm{dom}(S)$" is used to denote the domain of the set $S$.

Observe that if $S$ is a sequence in $X$ then $(S, \leq)$ is a net where the binary relation $\leq$ is defined such that for any two terms $S(m), S(n)$ in the sequence we have that $S(m) \leq S(n)$ if and only if $m$ is less than or equal to $n$ (in the number sense).

If $D$ denotes the domain of $S$, then it is common to use the notation to denote a net $(S, \leq)$:

(1)
\begin{align} \quad (S, \leq) = \{ S_n : n \in D, \leq \} \end{align}
 Definition: Let $X$ be a nonempty set and let $(S, \leq) = \{ S_n : n \in D, \leq \}$ be a net in $X$. Let $A \subseteq X$. Then the net $(S, \leq)$ is said to Eventually be in $A$ if there exists an $N \in D$ such that if $N \leq n$ then $S_n \in A$.