  We prove several topological properties of linear Weingarten surfaces of
Bryant type, as wave fronts in hyperbolic 3-space. For example, we show the
orientability of such surfaces, and also co-orientability when they are not
flat. Moreover, we show an explicit formula of the non-holomorphic hyperbolic
Gauss map via another hyperbolic Gauss map which is holomorphic. Using this, we
show the orientability and co-orientability of CMC-1 faces (i.e., constant mean
curvature one surfaces with admissible singular points) in de Sitter 3-space.
(CMC-1 faces might not be wave fronts in general, but belong to a class of
linear Weingarten surfaces with singular points.) Since both linear Weingarten
fronts and CMC-1 faces may have singular points, orientability and
co-orientability are both nontrivial properties. We also remark on some
properties of non-orientable maximal surfaces in Lorentz-Minkowski 3-space,
comparing the corresponding properties of CMC-1 faces in de Sitter 3-space.
