A necessary and sufficient condition for coincidence with the weak topology

Abstract

For a topological space $X$, it is a natural undertaking to compare its topology with the weak topology generated by a family of real-valued continuous functions on $X$. We present a necessary and sufficient condition for the coincidence of these topologies for an arbitrary family $\mathcal{A}\subset C\left(X\right)$. As a corollary, we give a new proof of the fact that families of functions which separate points on a compact space induce topologies that coincide with the original topology.