The relationship between pointwise k-pseudo metrics and pointwise k-closed ball systems

In this paper, we introduce a definition of a pointwise k-pseudo metric which is a generalization of pointwise pseudo metric. Then we discuss the axiomatic characterizations of a pointwise k-pseudo metric by a pointwise k-closed ball system that can be regarded as the opposite of the corresponding pointwise k-open ball system. Moreover, it is proved that there is a one-to-one correspondence between pointwise k-pseudo metrics and symmetric pointwise k-closed ball systems, and the symmetry conditions are different and simpler than the previous symmetry conditions.Finally, some L-structures (L denotes a completely distributive De Morgan algebra) induced by a pointwise k-closed ball system are obtained, including an L-neighborhood system, an L-interior operator and an L-topology.