Definition:Convergent Net/Cluster Point

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.

Let $x \in X$.

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


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

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 $\family {x_\lambda}_{\lambda \in \Lambda}$.