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.Comment: 36 pages, 1 figure, 8 diagrams. arXiv admin note: text overlap with arXiv:1607.04721

@article{paperbot819,
  title = {Web spaces and worldwide web spaces: topological aspects of domain theory},
  author = {Marcel Erné},
  journal = {Logical Methods in Computer Science},
  volume = {Volume 15, Issue 1},
  year = {2019},
  doi = {10.23638/lmcs-15(1:23)2019}
}