Let \((x_\alpha)_{\alpha \in J}\) be a net in \(X\). We say that \(x\) is an accumulation point of the net \((x_\alpha)\) if for each neighborhood \(U\) of \(x\), the set of those \(\alpha\) for which \(x_\alpha \in U\) is cofinal in \(J\).
Lemma. The net \((x_\alpha)\) has the point \(x\) as an accumulation point if and only if some subnet of \((x_\alpha)\) converges to \(x\).
[Hint: To prove the implication \(\Rightarrow\), let \(K\) be the set of all pairs \((\alpha, U)\) where \(\alpha \in J\) and \(U\) is a neighborhood of \(x\) containing \(x_\alpha\). Define \((\alpha, U) \preceq (\beta, V)\) if \(\alpha \preceq \beta\) and \(V \subset U\). Show that \(K\) is a directed set and use it to define the subnet.]
Solution
Suppose first that \(x\) is an accumulation point of the net \((x_\alpha)_{\alpha\in J}\). Let \(K\) be the set of pairs \((\alpha,U)\) such that \(U\) is a neighborhood of \(x\) and \(x_\alpha\in U\). Define \[ (\alpha,U)\preceq(\beta,V) \iff \alpha\preceq\beta\text{ and }V\subset U. \] Given two pairs, choose \(\gamma\succeq\alpha,\beta\). Since the indices with \(x_\delta\in U\cap V\) are cofinal, choose \(\delta\succeq\gamma\) with \(x_\delta\in U\cap V\). Then \((\delta,U\cap V)\) is an upper bound, so \(K\) is directed.
Define \(\phi:K\to J\) by \(\phi(\alpha,U)=\alpha\). It is order preserving and cofinal: for every \(\alpha_0\in J\), choose any neighborhood \(U\) of \(x\), then use accumulation to find \(\alpha\succeq\alpha_0\) with \(x_\alpha\in U\). Thus \((x_{\phi(k)})_{k\in K}\) is a subnet. It converges to \(x\): if \(W\) is a neighborhood of \(x\), choose \(k_0=(\alpha,W)\in K\); whenever \(k\succeq k_0\), its neighborhood component is contained in \(W\), hence the corresponding net point lies in \(W\).
Conversely, if a subnet converges to \(x\), then for every neighborhood \(U\) of \(x\) and every \(\alpha_0\in J\), eventually the subnet lies in \(U\), and cofinality of the subnet map gives an original index \(\alpha\succeq\alpha_0\) with \(x_\alpha\in U\). Hence the set of such indices is cofinal, so \(x\) is an accumulation point.