Definition:Convergent Net

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Definition

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

Let $\struct {I, \le}$ be a directed set.

Let $\family {x_i}_{i \mathop \in I}$ be a net.


$\family {x_i}$ is said to converge to $x \in X$, denoted $\ds x_i \to x$ or $\lim x_i = x$, if and only if:

$\forall U \in \tau: x \in U \implies \exists i_0 \in I: \forall i \ge i_0: x_i \in U$

That is, for every open $U$ with $x \in U$, there exists an $i_0 \in I$ such that forall $i \ge i_0$, $x_i \in U$.


If $x_i \to x$, then $x$ is called a limit (point) of $\family {x_i}$.


A net $\family {x_i}_{i \mathop \in I}$ is called convergent if there is an $x \in X$ such that $x_i \to x$.

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


Cluster Point

The net $\family {x_i}$ is said to cluster at $x \in X$, denoted $x_i \mathop {\longrightarrow_{\text{cl} } } x$, if and only if:

$\forall U \in \tau, i_0 \in I: x \in U \implies \exists i \ge i_0: x_i \in U$

That is, for every open $U$ with $x \in U$, and for every $i_0 \in I$, there is an $i \ge i_0$ such that $x_i \in U$.


If $x_i \mathop {\longrightarrow_{\text{cl} } } x$, then $x$ is called a cluster point of $\family {x_i}$.


Also see


Sources