  We construct a space $\mathbb{P}$ for which the canonical homomorphism
$\pi_1(\mathbb{P},p) \rightarrow \check{\pi}_1(\mathbb{P},p)$ from the
fundamental group to the first \v{C}ech homotopy group is not injective,
although it has all of the following properties: (1) $\mathbb{P}\setminus\{p\}$
is a 2-manifold with connected non-compact boundary; (2) $\mathbb{P}$ is
connected and locally path connected; (3) $\mathbb{P}$ is strongly
homotopically Hausdorff; (4) $\mathbb{P}$ is homotopically path Hausdorff; (5)
$\mathbb{P}$ is 1-UV$_0$; (6) $\mathbb{P}$ admits a simply connected
generalized covering space with monodromies between fibers that have discrete
graphs; (7) $\pi_1(\mathbb{P},p)$ naturally injects into the inverse limit of
finitely generated free monoids otherwise associated with the Hawaiian Earring;
(8) $\pi_1(\mathbb{P},p)$ is locally free.
