  We construct an example of a Peano continuum $X$ such that: (i) $X$ is a
one-point compactification of a polyhedron; (ii) $X$ is weakly homotopy
equivalent to a point (i.e. $\pi_n(X)$ is trivial for all $n \geq 0$); (iii)
$X$ is noncontractible; and (iv) $X$ is homologically and cohomologically
locally connected (i.e. $X$ is a $HLC$ and $clc$ space). We also prove that all
classical homology groups (singular, \v{C}ech, and Borel-Moore), all classical
cohomology groups (singular and \v{C}ech), and all finite-dimensional Hawaiian
groups of $X$ are trivial.
