Think of it this way de Sitter space is essentially analogous to living on a sphere. In that space any triangle on the sphere's surface has more than 180 degrees.
http://en.wikipedia.org/wiki/Spherical_geometry
In hyperbolic space any triangle on the hyperbolic surface has less than 180 degrees.
A sphere has no boundary (think invisible walls in video games). You can move over it for infinite time without encountering an invisible wall that says, you can't go further. Because it closes on itself. Contrast this with hyperbolic space, which must has such boundary, because it can't close on itself without becoming some other kind of space.
Anti de Sitter space is essentially living on a boundary of a black holes like curvature. Like this one: http://en.wikipedia.org/wiki/File:HyperboloidOfOneSheet.png
In hyperbolic space any triangle on the hyperbolic surface has less than 180 degrees.
A sphere has no boundary (think invisible walls in video games). You can move over it for infinite time without encountering an invisible wall that says, you can't go further. Because it closes on itself. Contrast this with hyperbolic space, which must has such boundary, because it can't close on itself without becoming some other kind of space.
Note: it's an analogy, don't be too nitpicky.