  The first author studied spacelike constant mean curvature one (CMC-1)
surfaces in de Sitter 3-space when the surfaces have no singularities except
within some compact subset and are of finite total curvature on the complement
of this compact subset. However, there are many CMC-1 surfaces whose singular
sets are not compact. In fact, such examples have already appeared in the
construction of trinoids given by Lee and the last author via hypergeometric
functions. In this paper, we improve the Osserman-type inequality given by the
first author. Moreover, we shall develop a fundamental framework that allows
the singular set to be non-compact, and then will use it to investigate the
global behavior of CMC-1 surfaces.
