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 subset of R-3 with the following properties: (1) There is a universal covering projection q : (T) over bar -> T with uncountable fundamental group pi(1) ((T) over bar); (2) For every 1 not equal [(alpha) over bar] is an element of pi(1) (T, *), there is a covering projection r : (E, e ) -> ((T) over bar, *) such that [(alpha) over bar] is not an element of r(#)pi(1) (E, e ); (3) There is no universal covering projection r : E -> (T) over bar ; (4) The universal object p : (T) over tilde -> T in the category of fibrations with unique path lifting (and path-connected total space) over T has trivial fundamental group pi(1) ((T) over tilde) = 1; (5) p : (T) over tilde -> T is not a path component of an inverse limit of covering projections over T.