  Trapezoidal words are words having at most $n+1$ distinct factors of length
$n$ for every $n\ge 0$. They therefore encompass finite Sturmian words. We give
combinatorial characterizations of trapezoidal words and exhibit a formula for
their enumeration. We then separate trapezoidal words into two disjoint
classes: open and closed. A trapezoidal word is closed if it has a factor that
occurs only as a prefix and as a suffix; otherwise it is open. We investigate
open and closed trapezoidal words, in relation with their special factors. We
prove that Sturmian palindromes are closed trapezoidal words and that a closed
trapezoidal word is a Sturmian palindrome if and only if its longest repeated
prefix is a palindrome. We also define a new class of words, \emph{semicentral
words}, and show that they are characterized by the property that they can be
written as $uxyu$, for a central word $u$ and two different letters $x,y$.
Finally, we investigate the prefixes of the Fibonacci word with respect to the
property of being open or closed trapezoidal words, and show that the sequence
of open and closed prefixes of the Fibonacci word follows the Fibonacci
sequence.
