From Monotone Functions to Scattered Orders

Gadi Moran, University of Haifa

We shall review the evolvement of some characterisations of Scattered Compact Ordered Spaces (SCOSs) prompted by a search for a generalisation of Lebesgue's structure theorem for real monotone functions on the unit interval (stating that every such function is (uniquely) the sum of a CONTINUOUS monotone function and its "jumps") in the realm of regulated functions - those real functions on the unit interval possesing the one-sided limits everywhere.

Here are three of these, the first a topological one, the other two via properties of the space of continuous functions C(K) over a compact Hausdorff space K:

1.The Following Are Equivalent for a compact Hausdorff space K: (a)K is a SCOS (endowed with a suitable linear order). (b)K is a two to one continuous image of a successor ordinal (i.e, a well ordered compact space).

2.TFAE for a compact ordered space K: (a)K is scattered (i.e, a SCOS). (b)Every continuous function on K is a sum of its increments. (c)Every countable set of jump-functions with distinct jump-locations in C(K) has at most one sum.

Another outcome of this work is an extension of Riemann's Theorem about the sumset of a conditionally convergent (cc) series of reals (that under rearrangements of its terms it will converge to any real number), as well as of its generalisation by E. Steinitz and P. Levy for cc series in any Euclidean space.