A simply connected universal fibration with unique path lifting over a Peano continuum with non-simply connected universal covering space

We present a 2-dimensional Peano continuum T ⊆ R 3 \mathbb {T}\subseteq \mathbb {R}^3 with the following properties: (1) There is a universal covering projection q : T ¯ → T q:\overline {\mathbb {T}}\rightarrow \mathbb {T} with uncountable fundamental group π 1 ( T ¯ ) \pi _1(\overline {\mathbb {T}}) ; (2) For every 1 ≠ [ α ¯ ] ∈ π 1 ( T ¯ , ∗ ) 1\not =[\overline {\alpha }]\in \pi _1(\overline {\mathbb {T}},\ast ) , there is a covering projection r : ( E , e ) → ( T ¯ , ∗ ) r:(E,e)\rightarrow (\overline {\mathbb {T}},\ast ) such that [ α ¯ ] ∉ r # π 1 ( E , e ) [\overline {\alpha }]\not \in r_\#\pi _1(E,e) ; (3) There is no universal covering projection r : E → T ¯ r:E\rightarrow \overline {\mathbb {T}} ; (4) The universal object p : T ~ → T p:\widetilde {\mathbb {T}}\rightarrow \mathbb {T} in the category of fibrations with unique path lifting (and path-connected total space) over T \mathbb {T} has trivial fundamental group π 1 ( T ~ ) = 1 \pi _1(\widetilde {\mathbb {T}})=1 ; (5) p : T ~ → T p:\widetilde {\mathbb {T}}\rightarrow \mathbb {T} is not a path component of an inverse limit of covering projections over T \mathbb {T} .