2019-10 Is there canonical topology for topological groups?

Let \(G\) be a group. A topology on \(G\) is said to be a group topology if the map \(\mu: G \times G \to G\) defined by \(\mu(g, h) = g^{-1}h\) is continuous with respect to this topology where \(G \times G\) is equipped with the product topology. A group equipped with a group topology is called a topological group. When we have two topologies \(T_1, T_2\) on a set S, we write \(T_1 \leq T_2\) if \(T_2\) is finer than \(T_1\), which gives a partial order on the set of topologies on a given set. Prove or disprove the following statement: for a give group \(G\), there exists a unique minimal group topology on \(G\) (minimal with respect to the partial order we described above) so that \(G\) is a Hausdorff space?

GD Star Rating
loading...
2019-10 Is there canonical topology for topological groups?, 3.6 out of 5 based on 7 ratings
This entry was posted in problem on by .

About Hyungryul

2003.3-2009.8 KAIST, Undergraduate student in Mathematics 2009.8-2014.8 Cornell University, PhD student in Mathematics 2014.9-2017.2 University of Bonn, Postdoc 2019.3-Present KAIST, Assistant Professor

One thought on “2019-10 Is there canonical topology for topological groups?

  1. LJH

    Is there any further assumption for a group topology on $G$ such as $T_1$ separation axiom or Hausdorff property?

Comments are closed.