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 $11.3$