  We introduce topological phases of matter defined by skyrmions in the ground
state spin -- or pseudospin -- expectation value textures in the Brillouin
zone, the chiral and helical topological skyrmion phases of matter. These
phases are protected by a symmetry present in centrosymmetric superconductors.
We consider a tight-binding model for spin-triplet superconductivity in
transition metal oxides and find it realizes each of these topological skyrmion
phases. The chiral phase is furthermore realized for a parameter set
characterizing Sr$_2$RuO$_4$ with spin-triplet superconductivity. We also find
two types of topological phase transitions by which the skyrmion number can
change. The second type occurs without the closing of energy gaps in a system
described by a quadratic Hamiltonian without breaking the protecting symmetries
when atomic spin-orbit coupling is non-negligible and there is a suitable
additional degree of freedom. This contradicts the ``flat band'' limit
assumption important in use of entanglement spectrum and Wilson loops, and in
construction of the ten-fold way classification scheme of topological phases of
matter. We furthermore predict two kinds of bulk-boundary correspondence
signatures -- one for measurements which execute a partial trace over degrees
of freedom other than spin, which yields quantized transport signatures -- and
a second resulting from skyrmions trapping defects with their own non-trivial
topology that is discussed in a second work, which yields generalizations of
unpaired Majorana zero-modes.
