Let \(X\) be a topological space. A net in \(X\) is a function \(f\) from a directed set \(J\) into \(X\). If \(\alpha \in J\), we usually denote \(f(\alpha)\) by \(x_\alpha\). We denote the net \(f\) itself by the symbol \((x_\alpha)_{\alpha \in J}\), or merely by \((x_\alpha)\) if the index set is understood.
The net \((x_\alpha)\) is said to converge to the point \(x\) of \(X\) (written \(x_\alpha \to x\)) if for each neighborhood \(U\) of \(x\), there exists \(\alpha \in J\) such that
\[ \alpha \preceq \beta \implies x_\beta \in U. \]
Show that these definitions reduce to familiar ones when \(J = \mathbb{Z}_+\).
Solution
Goal: Prove that when the directed set \(J\) is the set of positive integers \(\mathbb{Z}_+\) equipped with the standard ordering \(\le\), the definition of a net and net convergence in a topological space \(X\) coincides precisely with the classical definition of a sequence and sequence convergence.
1. \(\mathbb{Z}_+\) is a directed set under the standard ordering \(\le\):Proof:
1.1. The standard relation \(\le\) on \(\mathbb{Z}_+\) is reflexive (\(n \le n\)) and transitive (\(m \le n \land n \le k \implies m \le k\)).
1.2. For any two elements \(m, n \in \mathbb{Z}_+\), choose \(k = \max\{m, n\} \in \mathbb{Z}_+\). Then \(m \le k\) and \(n \le k\).
1.3. Therefore \((\mathbb{Z}_+, \le)\) is a directed set.
2. A net indexed by \(\mathbb{Z}_+\) is a sequence:Proof:
2.1. A net in \(X\) indexed by \(\mathbb{Z}_+\) is a function \(f: \mathbb{Z}_+ \to X\).
2.2. By standard definition, a sequence in \(X\) is a function from \(\mathbb{Z}_+\) into \(X\), written as \((x_n)_{n \in \mathbb{Z}_+}\) where \(x_n = f(n)\).
2.3. Thus nets indexed by \(\mathbb{Z}_+\) and sequences in \(X\) are the exact same objects.
3. Net convergence on \(\mathbb{Z}_+\) is sequence convergence:Proof:
3.1. Under the net convergence definition with \((J, \preceq) = (\mathbb{Z}_+, \le)\), \((x_n) \to x\) if and only if for every neighborhood \(U\) of \(x\), there exists \(N \in \mathbb{Z}_+\) such that:\[N \le n \implies x_n \in U.\]
3.2. Under the classical definition of sequence convergence in a topological space, \((x_n) \to x\) if and only if for every neighborhood \(U\) of \(x\), there exists an integer \(N\) such that \(x_n \in U\) for all \(n \ge N\).
3.3. These two conditions are logically and syntactically identical.
4. Conclusion:The theory of nets on the directed set \((\mathbb{Z}_+, \le)\) reduces directly to the standard theory of sequences. Q.E.D.