Lawvere Completeness in Topology

It is known since 1973 that Lawvere’s notion of (Cauchy-)complete enriched category is meaningful for metric spaces: it captures exactly Cauchy-complete metric spaces. In this paper we introduce the corresponding notion of Lawvere completeness for (T,V)-categories and show that it has an interesting meaning for topological spaces and quasi-uniform spaces: for the former ones means weak sobriety while for the latter means Cauchy completeness. Further, we show that V has a canonical (T,V)-category structure which plays a key role: it is Lawvere-complete under reasonable conditions on the setting; permits us to define a Yoneda embedding in the realm of (T,V)-categories. Mathematics Subject Classification (2000): 18A05, 18D15, 18D20, 18B35, 18C15, 54E15, 54E50.