# Definition:Convergent Net

This page has been identified as a candidate for refactoring of basic complexity.Until this has been finished, please leave
`{{Refactor}}` in the code.
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. |

## Definition

Let $\struct {X, \tau}$ be a topological space.

Let $\struct {I, \le}$ be a directed set.

Let $\family {x_i}_{i \mathop \in I}$ be a net.

$\family {x_i}$ is said to **converge to** $x \in X$, denoted $\ds x_i \to x$ or $\lim x_i = x$, if and only if:

- $\forall U \in \tau: x \in U \implies \exists i_0 \in I: \forall i \ge i_0: x_i \in U$

That is, for every open $U$ with $x \in U$, there exists an $i_0 \in I$ such that forall $i \ge i_0$, $x_i \in U$.

If $x_i \to x$, then $x$ is called a **limit (point) of $\family {x_i}$**.

A net $\family {x_i}_{i \mathop \in I}$ is called **convergent** if there is an $x \in X$ such that $x_i \to x$.

If such an $x$ does not exist, the net is said to be **divergent**.

### Cluster Point

The net $\family {x_i}$ is said to **cluster at** $x \in X$, denoted $x_i \mathop {\longrightarrow_{\text{cl} } } x$, if and only if:

- $\forall U \in \tau, i_0 \in I: x \in U \implies \exists i \ge i_0: x_i \in U$

That is, for every open $U$ with $x \in U$, and for every $i_0 \in I$, there is an $i \ge i_0$ such that $x_i \in U$.

If $x_i \mathop {\longrightarrow_{\text{cl} } } x$, then $x$ is called a **cluster point of $\family {x_i}$**.

## Also see

## Sources

- 1990: John B. Conway:
*A Course in Functional Analysis*(2nd ed.) Appendix $\text A.2.1$