  Monolayer transition metal dichalcogenides in the T' phase promise to realize
the quantum spin Hall (QSH) effect at room temperature, because they exhibit a
prominent spin-orbit gap between inverted bands in the bulk. Here we show that
the binding energy of electron-hole pairs excited through this gap is larger
than the gap itself in MoS2, a paradigmatic material that we investigate from
first principles by many-body perturbation theory (MBPT). This paradoxical
result hints at the instability of the T' phase against the spontaneous
generation of excitons, and indeed we find that it gives rise to a recostructed
`excitonic insulator' ground state. Importantly, we show that in this system
topological and excitonic order cooperatively enhance the bulk gap by breaking
the crystal inversion symmetry, in contrast to the case of bilayers where the
frustration between the two orders is relieved by breaking time reversal
symmetry. The excitonic topological insulator departs distinctively from the
bare topological phase as it lifts the band spin degeneracy, which results in
circular dichroism. A moderate biaxial strain applied to the system leads to
two additional excitonic phases, different in their topological character but
both ferroelectric as an effect of electron-electron interactions.
