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Abstract  We construct a space   P   for which the canonical homomorphismπ  1    (  P  ,  p  )   →    π  ̌   1    (  P  ,  p  )from the fundamental group to the first Čech homotopy group is not injective, although it has all of the following properties: (1)    P  ∖  {  p  }    is a 2‐manifold with connected non‐compact boundary; (2)   P   is connected and locally path connected; (3)   P   is strongly homotopically Hausdorff; (4)   P   is homotopically path Hausdorff; (5)   P   is 1‐UV0   ; (6)   P   admits a simply connected generalized covering space with monodromies between fibers that have discrete graphs; (7)π  1    (  P  ,  p  )naturally injects into the inverse limit of finitely generated free monoids otherwise associated with the Hawaiian Earring; (8)π  1    (  P  ,  p  )is locally free.

On the failure of the first Čech homotopy group to register geometrically relevant fundamental group elements