# Definition:Convergent Net

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## Definition

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

Let $\struct {\Lambda, \preceq}$ be a directed set.

Let $\family {x_\lambda}_{\lambda \mathop \in \Lambda}$ be a net.

We say that $\family {x_\lambda}_{\lambda \in \Lambda}$ **converges to** $x \in X$ in $\struct {X, \tau}$, denoted $\ds x_\lambda \to x$ or $\lim x_\lambda = x$, if and only if:

- $\forall U \in \tau: x \in U \implies \exists \lambda_0 \in \Lambda: \forall \lambda \succeq \lambda_0: x_\lambda \in U$

That is, for every open $U$ with $x \in U$, there exists an $\lambda_0 \in \Lambda$ such that forall $\lambda \succeq \lambda_0$, $x_\lambda \in U$.

A net $\family {x_\lambda}_{\Lambda \mathop \in \Lambda}$ is called **convergent** if there is an $x \in X$ such that $x_\lambda \to x$.

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

### Limit of Net

$x \in X$ is called a **limit (point) of $\family {x_\lambda}_{\lambda \in \Lambda}$** if and only if $\family {x_\lambda}_{\lambda \in \Lambda}$ converges to $x$.

### Cluster Point

We say that $\family {x_\lambda}_{\lambda \mathop \in \Lambda}$ **clusters at** $x \in X$, denoted $x_\lambda \mathop {\longrightarrow_{\text{cl} } } x$, if and only if:

- $\forall U \in \tau, \lambda_0 \in \Lambda: x \in U \implies \exists \lambda \succeq \lambda_0: x_\lambda \in U$

That is, for every open $U$ with $x \in U$, and for every $\lambda_0 \in \Lambda$, there is an $\lambda \ge \lambda_0$ such that $\lambda \in U$.

If $x_\lambda \mathop {\longrightarrow_{\text{cl} } } x$, then $x$ is called a **cluster point of $\family {x_\lambda}_{\lambda \in \Lambda}$**.

## Also see

## Sources

- 1990: John B. Conway:
*A Course in Functional Analysis*(2nd ed.) Appendix $\text A.2.1$ - 1970: Stephen Willard:
*General Topology*... (previous) ... (next): Chapter $4$: Convergence: $\S11$: Nets: Definition $11.3$