A necessary and sufficient condition for coincidence with the weak topology

Clanin, Joseph; Lee, Kristopher

doi:10.2140/involve.2017.10.257

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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

DOI: 10.2140/involve.2017.10.257

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 $A \subset C (X)$ . 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.

Keywords

weak topology, continuous functions

Mathematical Subject Classification 2010

Primary: 46E25, 54A10

Milestones

Received: 10 September 2015

Revised: 9 December 2015

Accepted: 19 December 2015

Published: 10 November 2016

Communicated by Joel Foisy

Authors

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

Vol. 10, No. 2, 2017