20 Sept 2004 

2:30 - 4:00   Gadi Moran 

From Monotone Functions to Scattered Orders 


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 intervals equipped with the
one sided limit 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.

An extension of Riemann's Theorem, that the sumset of a conditionally
convergent (cc) series of reals (under rearrangements of its terms) is
actually the set of all reals, as wellas of its generalisation by
E. Steinitz and P. Levy for cc series in any Euclidean space, occur also
in this context and will be discussed.