A definition that equates a hydrogen bond topologically, with a local energy well in the potential energy surface is used to study the structure and dynamics of liquid water. We demonstrate the robustness of this hydrogen-bond definition versus the many other definitions which use fixed, arbitrary parameters, do not account for variable molecular environments, and cannot effectively resolve transition states. Our topology definition unambiguously shows that most water molecules are double acceptors but sizable proportions are single or triple acceptors. Almost all hydrogens are found to take part in hydrogen bonds. Broken hydrogen bonds only form when two molecules try to form two hydrogen bonds between them. The double acceptors have tetrahedral geometry, lower potential energy, entropy, and density, and slower dynamics. The single and triple acceptors have trigonal and trigonal bipyramidal geometry and when considered together have higher density, potential energy, and entropy, faster dynamics, and a tendency to cluster. These calculations use an extended theory for the entropy of liquid water that takes into account the variable number of hydrogen bonds. Hydrogen-bond switching is shown to depend explicitly on the variable number of hydrogen bonds accepted and the presence of interstitial water molecules. Transition state theory indicates that the switching of hydrogen bonds is a mildly activated process, requiring only a moderate distortion of hydrogen bonds. Three main types of switching events are observed depending on whether the donor and acceptor are already sharing a hydrogen bond. The switch may proceed with no intermediate or via a bifurcated-oxygen or cyclic dimer, both of which have a broken hydrogen bond and symmetric and asymmetric forms. Switching is found to be strongly coupled to whole-molecule vibration, particularly for the more mobile single and triple acceptors. Our analysis suggests that even though water is heterogeneous in terms of the number of hydrogen bonds, the coupling between neighbors on various length and time scales brings about greater continuity in its properties.