Fuzzy topological spaces with conical neighborhood systems

For a commutative, integral, and meet continuous quantale Q, a CNS space is defined to be a Q-topological space such that the Q-neighborhood system of each point is a conical Q-filter. These spaces reduce to fuzzy T-locality spaces in the case that the quantale is the unit interval endowed with a left continuous t-norm T; and reduce to probabilistic Q-topological spaces in the case that Q is an MV-algebra. For a continuous quantale Q, a necessary and sufficient condition is obtained for the category of CNS spaces to be simultaneously reflective and coreflective in the category of stratified Q-topological spaces. This result provides an interesting example of the interaction between properties of the quantale Q and that of Q-topological spaces.