The whisker topology on fundamental groups preserves products, yields a Hausdorff non-discrete non-abelian topological group for an infinite product of wedges of two circles, and is non-separable on the Hawaiian earring group.
Biss, The topological fundamental group and generalized covering spaces, Topol- ogy and its Applications 124 (2002), 355–371
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On The Whisker Topology
The whisker topology on fundamental groups preserves products, yields a Hausdorff non-discrete non-abelian topological group for an infinite product of wedges of two circles, and is non-separable on the Hawaiian earring group.