Proof. Assume that \(X\) is Hausdorff. Suppose, for a contradiction, that \(x,y\in X\) are distinct and \(\net x{\ga }A\) is a net in \(X\) that converges to both \(x\) and \(y\). Let \(U\) and \(V\) be disjoint and open in \(X\) with \(x\in U\) and \(y\in V\). There are \(\gb ,\gm \in A\) with \(x_{\ga }\in U\) for \(\ga \ge \gb \) and \(x_{\ga }\in V\) for \(\ga \ge \gm \). Let \(\gd \in A\) be such that \(\ga \le \gd \) and \(\gb \le \gd \). Then \(x_{\gd }\in U\cap V\), which is a contradiction.

Assume that every net in \(X\) converges to at most one point in \(X\). Let \(x,y\in X\) be distinct and suppose, for a contradiction, that \(U\cap V\neq \emp \) for any open \(U,V\) in \(X\) with \(x\in U\) and \(y\in V\). Let

\[ A:=\set {\ang {U,V}:U,V\text { are open nbhds of }x\text { and }y,\text { respectively}}. \]

Define \(\le \) on \(A\) by

\[ \ang {U,V}\le \ang {U’,V’} \]

if \(U’\sub U\) and \(V’\sub V\). Then \(A\) becomes a directed set. Let \(x_{\ga }\in U\cap V\) for every \(\ga :=\ang {U,V}\in A\). Then \(\net x{\ga }A\) converges to \(x\) since for any open nbhd \(U\) of \(x\) we have \(x_{\ga }\in U\) for \(\ga \ge \ang {U,X}\). Similarly, \(\net x{\ga }A\) converges to \(y\), which is a contradiction.  □

Proof. Assume that there is a net \(\net x{\ga }A\) in \(A\) that converges to \(x\). Let \(U\) be a nbhd of \(x\). Then there is \(\gb \in A\) such that \(x_{\ga }\in U\) for every \(\ga \ge \gb \). In particular, \(x_{\gb }\in U\) so \(U\cap A\neq \emp \). Thus \(x\in \ob A\).

Now assume that \(x\in \ob A\). Let \(D\) be the set of all nbhds of \(x\) directed by inverted inclusion. For each \(\ga \in D\) there is \(x_{\ga }\in \ga \cap A\). Then \(\net x{\ga }A\) converges to \(x\).  □

Proof. Assume that \(f\) is continuous and \(\net x{\ga }A\) is a net in \(X\) that converges to \(x\in X\). Let \(U\) be a nbhd of \(f\of x\) in \(Y\). Then \(f^{-1}\bof U\) is a nbhd of \(x\) in \(X\) so there is \(\gb \in A\) such that \(x_{\ga }\in f^{-1}\bof U\) for every \(\ga \ge \gb \). Thus \(f\of {x_{\ga }}\in U\) for every \(\ga \ge \gb \) so \(\cur {f\of {x_{\ga }}}_{\ga \in A}\) converges to \(f\of x\).

Now assume that for every net \(\net x{\ga }A\) in \(X\) that converges to \(x\in X\), the net \(\cur {f\of {x_{\ga }}}_{\ga \in A}\) converges to \(f\of x\). If \(B\sub X\), and \(x\in \ob B\), then there is a net \(\net x{\ga }A\) in \(X\) that converges to \(x\). Since \(\cur {f\of {x_{\ga }}}_{\ga \in A}\) converges to \(f\of x\), it follows that \(f\of x\in \ob {f\bof A}\). Since \(f\bof {\ob A}\sub \ob {f\bof A}\) for any \(A\sub X\), it follows that \(f\) is continuous.  □

Proof. Assume that \(\net x{\gb }B\) converges to \(x\). Since \(p_{\ga }\) is continuous, it follows that \(\cur {p_{\ga }\of {x_{\gb }}}_{\gb \in B}\) converges to \(p_{\ga }\of x\) for every \(\ga \in A\).

Assume that \(\cur {p_{\ga }\of {x_{\gb }}}_{\gb \in B}\) converges to \(p_{\ga }\of x\) for every \(\ga \in A\). Let \(U\) be a nbhd of \(x\). There is a finite \(A’\sub A\) and open \(U_{\ga }\) in \(X_{\ga }\) for every \(\ga \in A’\) such that

\[ \prod _{\ga \in A}U_{\ga }\sub U, \]

where \(U_{\ga }:=X_{\ga }\) for every \(\ga \in A\sem A’\). Let

\[ A’:=\set {\ga _{1},\ga _{2},\ds ,\ga _{n}}. \]

For each \(\ga \in A’\) there is \(\gb _{\ga }\in B\) such that \(p_{\ga }\of {x_{\gb }}\in U_{\ga }\) for every \(\gb \ge \gb _{\ga }\). Since \(A’\) is finite, there is \(\gm \in B\) such that \(\gb _{\ga }\le \gm \) for every \(\ga \in A’\). If \(\gb \ge \gm \), then \(p_{\ga }\of {x_{\gb }}\in U_{\ga }\) for every \(\ga \in A\) so \(\net x{\gb }B\) converges to \(x\).  □

Proof. Assume that there exists a subnet \(\net y{\gb }B\) of \(\net x{\ga }A\) that converges to \(x\). Let \(U\) be a nbhd of \(x\) in \(X\). There is \(\gb _{0}\in B\) such that \(y_{\gb }\in U\) for every \(\gb \ge \gb _{0}\). Let \(\ph :B\to A\) define the subnet. Let \(\ga \in A\). There is \(\gb _{1}\in B\) such that \(\ph \of {\gb }\ge \ga \) for every \(\gb \ge \gb _{1}\). Let \(\gb \in B\) be such that \(\gb \ge \gb _{0}\) and \(\gb \ge \gb _{1}\). then

\[ x_{\ph \of {\gb }}=y_{\gb }\in U \]

and \(\ph \of {\gb }\ge \ga \). Thus \(\net x{\ga }A\) is frequently in \(U\), which implies that \(x\) is a cluster point of \(\net x{\ga }A\).

Now assume that \(x\) is a cluster point of \(\net x{\ga }A\). Let \(B\) be the family of all pairs \(\ang {\ga ,U}\) with \(\ga \in A\) and \(U\) being a nbhd of \(x\) with \(x_{\ga }\in U\). Define a direction \(\le \) on \(B\) by

\[ \ang {\ga ,U}\le \ang {\ga ’,U’} \]

if \(\ga \le \ga ’\) and \(U’\sub U\). Then \(\ph :B\to A\) given by \(\ph \of {\ga ,U}:=\ga \) defines a subnet \(\net y{\gb }B\) of \(\net x{\ga }A\). We show that \(\net y{\gb }B\) converges to \(x\). Let \(U\) be a nbhd of \(x\). There is \(\ga \in A\) with \(x_{\ga }\in U\). Let \(\gb _{0}:=\ang {\ga ,U}\). If \(\gb :=\ang {\ga ’,U’}\ge \gb _{0}\), then

\[ y_{\gb }=x_{\ph \of {\gb }}=x_{\ga ’}\in U’\sub U, \]

as required.  □