Definition:Convergent Net/Cluster Point

It has been suggested that this page be renamed. In particular: and move it into its own space To discuss this page in more detail, feel free to use the talk page.

It has been suggested that this page or section be merged into all the other stuff. 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 {{Mergeto}} from the code.

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

This page may be the result of a refactoring operation. As such, the following source works, along with any process flow, will need to be reviewed. When this has been completed, the citation of that source work (if it is appropriate that it stay on this page) is to be placed above this message, into the usual chronological ordering. In particular: work out where Conway fits If you have access to any of these works, then you are invited to review this list, and make any necessary corrections. 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 {{SourceReview}} from the code.

• 1970: Stephen Willard: General Topology ... (previous) ... (next): Chapter $4$: Convergence: $\S11$: Nets: Definition $11.3$