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

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 A C(X). 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.

weak topology, continuous functions
Mathematical Subject Classification 2010
Primary: 46E25, 54A10
Received: 10 September 2015
Revised: 9 December 2015
Accepted: 19 December 2015
Published: 10 November 2016

Communicated by Joel Foisy
Joseph Clanin
Department of Mathematics
Iowa State University
Ames, IA 50014
United States
Kristopher Lee
Department of Mathematics
Iowa State University
Ames, IA 50014
United States