Convergence of Nets

Subjects: Topology
Links: Convergence of Filters, Convergence of Sequences, Directed Sets, Cofinal and Coinitial Subsets, Hausdorff Spaces

Def: A net on a set $X$ is a function $r:\mathrm{\Lambda }\to X$, where $\mathrm{\Lambda }$ is a directed set. To the point $r(\lambda )$ is denoted frequently as $x_{\lambda }$, and the expression "$r:\mathrm{\Lambda }\to X$ is a net" is also written as "$(x_{\lambda }{)}_{\lambda \in \mathrm{\Lambda }}$ is a net".

Obs: Every sequence on $X$ is a net on $X$.

Def: A point $x$ is called a limit of a net $(x_{\lambda }{)}_{\lambda \in \mathrm{\Lambda }}$ is for every neighbourhood $U$ of $x$ there exists a ${\lambda }_0\in \mathrm{\Lambda }$ such that $x_{\lambda }\in U$ for every $\lambda \ge {\lambda }_0$, we say that the net $(x_{\lambda }{)}_{\lambda \in \mathrm{\Lambda }}$ converges to $x$. A net can converge to many points; the set of all limit points of the net $(x_{\lambda }{)}_{\lambda \in \mathrm{\Lambda }}$ is denoted as $\underset{\lambda \in \mathrm{\Lambda }}{lim}{x}_{\lambda }$.

Def: A point $x$ is called a cluster point of a net $(x_{\lambda }{)}_{\lambda \in \mathrm{\Lambda }}$ if for every neighbourhood $U$ of $x$ and every ${\lambda }_0\in \mathrm{\Lambda }$ there exists a $\lambda \ge {\lambda }_0$ such that $x_{\lambda }\in U$.

Def: We say that a net $(x_{\sigma }{)}_{\sigma \in \mathrm{\Sigma }}$ is finer than the net $(x_{\lambda }{)}_{\lambda \in \mathrm{\Lambda }}$ if there is a function $\varphi :\mathrm{\Sigma }\to \mathrm{\Lambda }$ with the following properties:

• For every ${\lambda }_0\in \mathrm{\Lambda }$ there exists a ${\sigma }_0\in \mathrm{\Sigma }$ such that $\varphi (\sigma )\ge {\lambda }_0$ whenever $\sigma \ge {\sigma }_0$.
• $x_{\varphi (\sigma )}=x_{\sigma }$ for $\sigma \in \mathrm{\Sigma }$.
  We just make the observation that a non-decreasing function $\varphi :\mathrm{\Sigma }\to \mathrm{\Lambda }$ such that $\varphi [\mathrm{\Sigma }]$ is cofinal in $\mathrm{\Lambda }$.

Prop: If $x$ is a cluster point of the net $(x_{\sigma }{)}_{\sigma \in \mathrm{\Sigma }}$ that is finer than the net $(x_{\lambda }{)}_{\lambda \in \mathrm{\Lambda }}$, then $x$ is a cluster point of $(x_{\lambda }{)}_{\lambda \in \mathrm{\Lambda }}$. If $x$ is a limit point of the net $(x_{\lambda }{)}_{\lambda \in \mathrm{\Lambda }}$, then $x$ is also a limit point of every net $(x_{\sigma }{)}_{\sigma \in \mathrm{\Sigma }}$ finer than $(x_{\lambda }{)}_{\lambda \in \mathrm{\Lambda }}$. If $x$ is a cluster point of the net $(x_{\lambda }{)}_{\lambda \in \mathrm{\Lambda }}$, then $x$ is a limit point of a net $(x_{\sigma }{)}_{\sigma \in \mathrm{\Sigma }}$ that is finer than $(x_{\lambda }{)}_{\lambda \in \mathrm{\Lambda }}$.

Prop: Let $A$ be subset of a topological space $X$. For any $x\in X$, $x\in {\text{cl}}_X(A)$ iff there's a net $(x_{\lambda }{)}_{\lambda \in \mathrm{\Lambda }}$ on $A$ that converges to $x$.

Cor: A set $A$ is closed iff if together with any net it contains its limit points.

Cor: the point $x\in \text{Lim}(A)$ iff there exists a net $(x_{\lambda }{)}_{\lambda \in \mathrm{\Lambda }}$ that converges to $x$, such that $x_{\lambda }\in A$, and $x_{\lambda }\ne x$ for all $\lambda \in \mathrm{\Lambda }$.

Prop: Let $X$ and $Y$ be topological spaces, and $f:X\to Y$ be a function. $f$ is a continuous function iff $$f[\lim_{\lambda \in \Lambda} x_\lambda] \subseteq \lim_{\lambda \in \Lambda} f(x_\lambda)$$for every net $(x_{\lambda }{)}_{\lambda \in \mathrm{\Lambda }}$ in the space $X$.

Prop: A topological space $X$ is a $T_2$ space iff every net in $X$ has at most one limit.

Th: For every net $(x_{\lambda }{)}_{\lambda \in \mathrm{\Lambda }}$ in a topological space, the family $\mathcal{F}((x_{\lambda }{)}_{\lambda \in \mathrm{\Lambda }})$ consisting of all sets $A\subseteq X$ with the property that there exists a ${\lambda }_0\in \mathrm{\Lambda }$ such that $x_{\lambda }\in A$ whenever $\lambda \ge {\lambda }_0$ is f filter in the subspace and $lim\mathcal{F}((x_{\lambda }{)}_{\lambda \in \mathrm{\Lambda }})=\underset{\lambda \in {\mathrm{\Lambda }}_0}{lim}{x}_{\lambda }$. If the net $(x_{\sigma }{)}_{\sigma \in \mathrm{\Sigma }}$ is finer than the net $(x_{\lambda }{)}_{\lambda \in \mathrm{\Lambda }}$, then the filter $\mathcal{F}((x_{\sigma }{)}_{\sigma \in \mathrm{\Sigma }})$ is finer than the filter $\mathcal{F}((x_{\lambda }{)}_{\lambda \in \mathrm{\Lambda }})$.

Th: Let $\mathcal{F}$ be a filter in a topological space $X$; let us denote $\mathrm{\Lambda }$ the set of all pairs $(x,A)$, where $x\in A\in \mathcal{F}$ and let us define $(x_1,A_1)\le (x_2,A_2)$ if $A_2\subseteq A_1$. The set $\mathrm{\Lambda }$ is directed by $\le$, and for the net $(x_{\lambda }{)}_{\lambda \in \mathrm{\Lambda }}$ is defined as $x_{\lambda }=x$ for $\lambda =(x,A)\in \mathrm{\Lambda }$, we have that $\mathcal{F}=\mathcal{F}(\mathrm{\Lambda }(\mathcal{F}))$ and $\underset{\lambda \in \mathrm{\Lambda }}{lim}{x}_{\lambda }=lim\mathcal{F}$.

Th: A topological space $X$ is compact iff every net in $X$ has a cluster point.