Convergence of Sequences and Nets in Topological Spaces

Recall from the Sequences and Nets page that a sequence on a nonempty set $X$ is a function $S$ whose domain is the set of natural numbers, and that a net on $X$ is a pair $(S, \leq) = \{ S_n : n \in D, \leq \}$ where $S$ is a function on $X$ and $\leq$ is a binary relation that directs the domain $D = \mathrm{dom}(S)$. We said that a net is eventually in a set $A$ if there exists an $N \in D$ such that if $N \leq n$ then $S_n \in A$.

We are now ready to discuss the concepts of sequence convergence and net convergence in topological spaces.

Sequential Convergence in Topological Spaces

Definition: Let $(X, \tau)$ be a topological space. A sequence of points $(x_n) \subset X$ is said to Converge to a point $x \in X$ if for every open neighbourhood $B$ of $X$ there exists an $N \in \mathbb{N}$ such that if $n \geq N$ then $x_n \in B$.

Net Convergence in Topological Spaces

Definition: Let $(X, \tau)$ be a topological space. A net $(S, \leq) = \{ S_n : n \in D, \leq \}$ (where $D$ is the domain of the function $S$) is said to Converge to a point $s \in X$ if for every open neighbourhood $B$ of $X$ the net is eventually in $B$.

Sometimes the term "Moore-Smith Convergence is used in place of convergence for nets.

Example 1

Let $X$ be any nonempty set equipped with the indiscrete topology, that is, the only open sets are $\emptyset$ and $X$ itself. Let $(S, \leq) = \{ S_n : n \in D, \leq \}$ be a net in $X$. Observe that this net converges to every point in $X$. Indeed, if $s \in X$ then the only open neighbourhood of $s$ is $X$ itself, and certainly, $(S, \leq)$ is eventually in $X$.

In particular, if $X$ contains at least $2$ elements and is endowed with the indiscrete topology then any net in $X$ converges to every point in $X$ and so the convergence is not unique.

Example 2

Let $X$ be any nonempty set equipped with the discrete topology, that is, every subset of $X$ is open. Let $(S, \leq) = \{ S_n : n \in D, \leq \}$ be a net in $X$. Observe that since singleton sets are open, then the net $(S, \leq)$ converges to $s \in X$ if and only if $(S, \leq)$ is eventually in $\{ s \}$. In other words, $(S, \leq)$ converges to $s$ if and only if there exists an $N \in D$ such that $S_n = s$ for all $n$ with $N \leq n$.