  We systematically examine the topological, valleytronic, and optical
properties of experimentally accessible PbS (001) few-layers, with a focus on
the monolayer. With even-odd layer-dependent oscillations and without band
inversions, the band gaps cover a wide spectrum from infrared to visible,
making the few-layers promising for optoelectronics. Intriguingly, the uniaxial
(biaxial) compressive strain can tune the trivial monolayer into a
$\mathcal{Z}_2$-topological (topological crystalline) insulator, which may be
utilized for controllable low-power electronic devices. Although elliptical
dichroism vanishes in the monolayer due to inversion symmetry, optical pumping
provides an efficient tool to characterize the three phases and to realize
charge, spin, and valley Hall effects in optoelectronic transport that are
tunable by external strain and light ellipticity.
