IS THE CONSTRUCT OF L-TOPOLOGICAL SPACES A CO-TOWER EXTENSION OF SOME SIMPLER CONSTRUCT?

Abstract

In an earlier paper, the second author associated with any fibre-small topological construct A and any completely distributive lattice L, an extension of A, called the (L-)-co-tower extension of A. He demonstrated that many familiar constructs in fuzzy topology can be expressed as co-tower extensions of more basic constructs and asked whether the construct L-Top of L-topological spaces (the stratified Chang-Goguen spaces) can be expressed as a co-tower extension. In this note we answer this question with “no” and “yes”. In particular we present the following theorems: (1) If L is a 3-element linearly ordered lattice (or, more generally, any linearly ordered finite lattice with at least 3 elements), then L-Top cannot be expressed as an L-co-tower extension. (2) If L is a 4-element Boolean algebra (or, more generally, any atomic complete Boolean algebra), then L-Top is (up to concrete isomorphism) the L-co-tower extension of the construct Top of topological spaces.