  Let C(K) denote the Banach algebra of continuous real functions, with the
supremum norm, on a compact Hausdorff space K. For two subsets of C(K), one can
define their product by pointwise multiplication, just as the Minkowski sum of
the sets is defined by pointwise addition. Our main interest is in correlations
between properties of the product of closed order intervals in C(K) and
properties of the underlying space K. When K is finite, the product of two
intervals in C(K) is always an interval. Surprisingly, the converse of this
result is true for a wide class of compacta. We show that a first-countable
space K is finite whenever it has the property that the product of two
nonnegative intervals is closed, or the property that the product of an
interval with itself is convex. That some assumption on K is needed, can be
seen from the fact that, if K is the Stone-Cech compactification of N, then the
product of two intervals in C(K) with continuous boundary functions is always
an interval. For any K, it is proved that the product of two positive intervals
is always an interval, and that the product of two nonnegative intervals is
always convex. Finally, square roots of intervals are investigated, with
results of similar type.
