l{N}_{x}\Leftrightarrow(\forall N\in\mathcal{N}_{x})(\forall F\in\mathcal{F}_{\chi})\textrm{ }(N\bigcap F\neq\emptyset)\Leftrightarrow x\in\textrm{adh}(\mathcal{F}_{\chi})$; here $F$ is a superset of $\chi(\mathbb{R}_{\alpha})$.

Some examples of convergence of filters are

\(1) Any filter on an indiscrete space $X$ converges to every point of $X$.

\(2) Any filter on a space that coincides with its topology (minus the empty set, of course) converges to every point of the space.

\(3) For each $x\in X$, the neighbourhood filter $\mathcal{N}_{x}$ converges to $x$; this is the smallest filter on $X$ that converges to $x$.

\(4) The *indiscrete* filter $\mathcal{F}=\{ X\}$ converges to no point in the space $(X,\{\emptyset,A,X-A,X\})$, but converges to every point of $X-A$ if $X$ has the topology $\{\emptyset,A,X\}$ because the only neighbourhood of any point in $X-A$ is $X$ which is contained in the filter.

One of the most significant consequences of convergence theory of sequences and nets, as shown by the two theorems and the corollary following, is that this can be used to describe the topology of a set. The proofs of the theorems also illustrate the close inter-relationship between nets and filters.

**Theorem A1.4.** *For a subset $A$ of a topological space $X$,* $$\textrm{Cl}(A)=\{ x\in X\!:(\exists\textrm{ a net }\chi\textrm{ in }A)\textrm{ }(\chi\rightarrow x)\}.\qquad\square\label{Eqn: net closure}$$ **Proof.** *Necessity.* For **$x\in\textrm{Cl}(A)$, construct a **net **$\chi\rightarrow x$ in *$A$* as **follows. Let $\mathcal{B}_{x}$ be a topological local base at $x$, which by definition is the collection of all open sets of $X$ containing $x$. For each $\beta\in\mathbb{D}$, the sets $$N_{\beta}=\bigcap_{\alpha\preceq\beta}\{ B_{\alpha}\!:B_{\alpha}\in\mathcal{B}_{x}\}$$ form a nested decreasing local neighbourhood filter base at $x$. With respect to the directed set $_{\mathbb{D}}N_{\beta}=\{(N_{\beta},\beta)\!:(\beta\in\mathbb{D})(x_{\beta}\in N_{\beta})\}$ of Eq. (\[Eqn: DirectedIndexed\]), define th