## Erné, Marcel - Web spaces and worldwide web spaces: topological aspects of domain theory

lmcs:1539 - Logical Methods in Computer Science, March 5, 2019, Volume 15, Issue 1
Web spaces and worldwide web spaces: topological aspects of domain theory

Authors: Erné, Marcel

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.

Source : oai:arXiv.org:1802.08170
DOI : 10.23638/LMCS-15(1:23)2019
Volume: Volume 15, Issue 1
Published on: March 5, 2019
Submitted on: July 20, 2016
Keywords: Mathematics - General Topology,primary: 06B35, secondary: 06D10, 54D10, 54D45, 54F05