Functional representation of algebraic intervals
Robert E. Jamison · Pacific Journal of Mathematics · 1974
Motivated by some examples from the study of axiomatic convexity, we define a class of objects (in real algebras with 1) whose algebraic properties mimic those of the unit interval.These objects, called intervals, have quite a bit of structure in themselves.In particular, in a Banach algebra a compact interval must be finite dimensional.Even more striking is the main result which shows that any interval satisfying a very modest boundedness condition is commutative and can be represented by continuous functions from a compact Hausdorff space into the unit interval.This leads to a number of corollaries in analysis and topology.