  The paper discusses the question of why intervals, which are the main object
of Interval Analysis, have exactly the form that we know well and habitually
use, and not some other. In particular, we investigate why traditional
intervals are closed, i.\,e. contain their endpoints, and also what is wrong
with an empty interval. The second question considered in the work is how
expedient it is to expand the set of traditional intervals by some other
objects. We show that improper ("reversed") intervals and the arithmetic of
such intervals (Kaucher complete interval arithmetic) are very useful from many
different points of view.
