The coexistence of charge density wave (CDW) and superconductivity in tantalum disulfide (2H-TaS$_2$) at ambient pressure, is boosted by applying hydrostatic pressures up to 30GPa, thereby inducing a typical dome-shaped superconducting phase. The ambient pressure CDW ground state which begins at TCDW = 76 K, with critically small Fermi surfaces, was found to be fully suppressed at Pc = 8.7GPa. Around Pc, we observe a superconducting dome with a maximum superconducting transition temperature Tc = 9.1 K. First-principles calculations of the electronic structure predict that, under ambient conditions, the undistorted structure is characterized by a phonon instability at finite momentum close to the experimental CDW wave vector. Upon compression, this instability is found to disappear, indicating the suppression of CDW order. The calculations reveal an electronic topological transition (ETT), which occurs before the suppression of the phonon instability, suggesting that the ETT alone is not directly causing the structural change in the system. The temperature dependence of the first vortex penetration field has been experimentally obtained by two independent methods and the corresponding lower critical field H$_{c1}$ was deduced. While a d wave and single-gap BCS prediction cannot describe our H$_{c1}$ experiments, the temperature dependence of the H$_{c1}$ can be well described by a single-gap anisotropic s-wave order parameter.
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