Vol. 10, No. 2, 2017

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A necessary and sufficient condition for coincidence with the weak topology

Joseph Clanin and Kristopher Lee

Vol. 10 (2017), No. 2, 257–261
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 $\mathsc{A}\subset C\left(X\right)$. 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.

Keywords
weak topology, continuous functions
Mathematical Subject Classification 2010
Primary: 46E25, 54A10