  We theoretically study superconductivity in UTe$_2$, which is a
recently-discovered strong candidate for an odd-parity spin-triplet
superconductor. Theoretical studies for this compound faced difficulty because
first-principles calculations predict an insulating electronic state,
incompatible with superconducting instability. To overcome this problem, we
take into account electron correlation effects by a GGA$+U$ method and show the
insulator-metal transition by Coulomb interaction. Using Fermi surfaces
obtained as a function of $U$, we clarify topological properties of possible
superconducting states. Fermi surface formulas for the three-dimensional
winding number and three two-dimensional $\mathbb{Z}_2$ numbers indicate
topological superconductivity at an intermediate $U$ for all the odd-parity
pairing symmetry in the $Immm$ space group. Symmetry and topology of
superconducting gap node are analyzed and the gap structure of UTe$_2$ is
predicted. Topologically protected low-energy excitations are highlighted, and
experiments by bulk and surface probes are proposed to link Fermi surfaces and
pairing symmetry. Based on the results, we also discuss multiple
superconducting phases under magnetic fields, which were implied by recent
experiments.
