Definition:Convergent Net/Cluster Point
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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.
Let $x \in X$.
We say that $\family {x_\lambda}_{\lambda \mathop \in \Lambda}$ clusters at $x \in X$, denoted $x_\lambda \mathop {\longrightarrow_{\text{cl} } } x$, 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 $\family {x_\lambda}_{\lambda \in \Lambda}$.
Sources
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- 1970: Stephen Willard: General Topology ... (previous) ... (next): Chapter $4$: Convergence: $\S11$: Nets: Definition $11.3$