Abstract
The core of a point in a topological space is the intersection of its neighborhoods. We construct certain completions and compactifications for densely core-generated spaces, i.e., T 0-spaces having a subbasis of open cores such that the points with open cores are dense in the associated patch space. All T 0-spaces with a minimal basis are in that class. Densely core-generated spaces admit not only a coarsest quasi-uniformity (the unique totally bounded transitive compatible quasi-uniformity), but also a purely order-theoretical description by means of their specialization order and a suitable join-dense subset (join-basis). It turns out that the underlying ordered sets of the completions and compactifications obtained are, up to isomorphism, certain ideal completions of the join-basis. The topology of the resulting completion or compactification is the Lawson topology or the Scott topology, or a slight modification of these.
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Erné, M. Ideal Completions and Compactifications. Applied Categorical Structures 9, 217–243 (2001). https://doi.org/10.1023/A:1011260817824
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DOI: https://doi.org/10.1023/A:1011260817824