Definition:Convergent Net
 | This page has been identified as a candidate for refactoring of basic complexity. Until this has been finished, please leave {{Refactor}} in the code.
New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only. Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.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 {{Refactor}} from the code. |
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
- 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) Appendix $\text A.2.1$