Definition:Limit of Net

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.

$x \in X$ is called a limit (point) of $\family {x_\lambda}_{\lambda \in \Lambda}$ if and only if $\family {x_\lambda}_{\lambda \in \Lambda}$ converges to $x$.

Also see

• Definition:Limit of Sequence

Sources

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• 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 $11.3$