  We study structural and thermophysical properties of a one-dimensional
classical fluid made of penetrable spheres interacting via an attractive
square-well potential. Penetrability of the spheres is enforced by reducing
from infinite to finite the repulsive energy barrier in the pair potentials As
a consequence, an exact analytical solution is lacking even in one dimension.
Building upon previous exact analytical work in the low-density limit [Santos
\textit{et al.}, Phys. Rev. E \text{77}, 051206 (2008)], we propose an
approximate theory valid at any density and in the low-penetrable regime. By
comparison with specialized Monte Carlo simulations and integral equation
theories, we assess the regime of validity of the theory. We investigate the
degree of inconsistency among the various routes to thermodynamics and explore
the possibility of a fluid-fluid transition. Finally we locate the dependence
of the Fisher-Widom line on the degree of penetrability. Our results constitute
the first systematic study of penetrable spheres with attractions as a
prototype model for soft systems.
