Abstract
For a topological space X (a), (b), (c), (d) are equivalent: (a) X is an essentially complete To-space. (b) (i) X is sober; (ii) X is an upper semi-lattice with o with regard to its induced partial order such that the binary sup: X × X → X is continuous. (c) (i) X is a To-space; (ii) X is a complete lattice in its induced partial order such that for every set I the I-indexed sup: XI → X is continuous (d) (i) For every ordinary proper filter F on X there is a unique x ∈ X with X=conv\(\dot x^ - _1 \). —The essential hull λ X of a To-space X can be constructed as a space
of “convergence sets”. —Suitably topologizing a poset (X, ≤) one obtains (i) Frink's ideal completion, (ii) MacNeille's completion as the lattice underlying
.
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Hoffmann, RE. Essentially complete TO-spaces. Manuscripta Math 27, 401–432 (1979). https://doi.org/10.1007/BF01507294
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DOI: https://doi.org/10.1007/BF01507294