Endowed with natural topologies, the fundamental group of the Hawaiian earring continuously injects into the inverse limit of free groups. This note shows the injection fails to have a continuous inverse. Such a phenomenon was unexpected and appears to contradict results of another author.
"The topological Hawaiian earring group does not embed in the inverse limit of free groups." Algebr. Geom. Topol. 5 (4) 1585 - 1587, 2005. https://doi.org/10.2140/agt.2005.5.1585