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

TL;DR

This paper explores the relationship between injective approach spaces and continuous lattices within an enriched [0,1]-context, extending classical order-topology results using continuous t-norms.

Contribution

It generalizes Scott's classical result to an enriched setting with [0,1]-approach spaces and continuous t-norms, revealing new insights into their structure.

Findings

Specialization [0,1]-order of injective approach spaces is a continuous [0,1]-lattice.

The [0,1]-approach structure matches the Scott [0,1]-approach structure of the specialization order.

The converse of the classical result does not hold in the enriched context.

Abstract

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.