A necessary and sufficient condition for coincidence with the weak topology (Q342635)

Given a topological space \((X,\tau)\), let \(C(X)\) be the collection of all continuous functions from \((X,\tau)\) to \(\mathbb{R}\), \(\mathbb{R}\) with its usual topology. It is known that \(\tau=\tau_{\mathcal{A}}\) when \(\mathcal{A}\subset C(X)\) is completely regular, where \(\tau_{\mathcal{A}}\) denotes the weak topology on \(X\) generated by \(\mathcal{A}\). In this paper the authors define a generalization of the notion of a completely regular family \(\mathcal{A}\subset C(X)\), called finitely completely regular family. Using this notion, the authors prove the following: \(\tau=\tau_{\mathcal{A}}\) if and only if \(\mathcal{A}\subset C(X)\) is finitely completely regular. From this result, the authors show that \(\tau=\tau_{\mathcal{A}}\) follows immediately when \((X,\tau)\) is compact and \(\mathcal{A}\) separates points. Also, interesting and illustrative examples are given.