  Web spaces, wide web spaces and worldwide web spaces (alias C-spaces) provide
useful generalizations of continuous domains. We present new characterizations
of such spaces and their patch spaces, obtained by joining the original
topology with a second topology having the dual specialization order; these
patch spaces possess good convexity and separation properties and determine the
original web spaces. The category of C-spaces is concretely isomorphic to the
category of fan spaces; these are certain quasi-ordered spaces having
neighborhood bases of fans, obtained by deleting a finite number of principal
dual ideals from a principal dual ideal. Our approach has useful consequences
for domain theory, because the T$_0$ web spaces are exactly the generalized
Scott spaces of locally approximating ideal extensions, and the T$_0$ C-spaces
are exactly the generalized Scott spaces of globally approximating and
interpolating ideal extensions. We extend the characterization of continuous
lattices as meet-continuous lattices with T$_2$ Lawson topology and the
Fundamental Theorem of Compact Semilattices to non-complete posets. Finally,
cardinal invariants like density and weight of the involved objects are
investigated.
