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