Continuous [0,1]-lattices and injective [0,1]-approach spaces

In 1972, Dana Scott proved a fundamental result on the connection between order and topology which says that injective T-0 spaces are precisely continuous lattices endowed with Scott topology. This paper investigates whether this is true in an enriched context, where the enrichment is the quantale obtained by equipping the interval [0, 1] with a continuous t-norm. It is shown that for each continuous t-norm, the specialization [0, 1]-order of a separated and injective [0, 1]-approach space X is a continuous [0, 1]-lattice and the [0, 1]-approach structure of X coincides with the Scott [0, 1]-approach structure of its specialization [0, 1]-order; but, unlike in the classical situation, the converse fails in general. (c) 2021 Elsevier B.V. All rights reserved.