Abstract
We can only agree with Peter Johnstone who wrote in Johnstone (Bull Amer Math Soc (N.S.) 8:41–53, 1983) that
the first person (apart of Stone) to exploit the possibility of applying lattice theory to topology was Henry Wallman.
In his article Wallman (Ann Math 39, 112–126, 1938) published in 1938 (already briefly mentioned in the Introduction), Wallman presented a compactification technically based on lattice theoretic principles, and proved that to determine the homology type of a space X one needs only the lattice of closed sets. When doing that, he needed a lattice formula substituting a sufficiently weak topological separation. His ingenious idea of the “disjunctive property”, namely the requirement that
if a ≠ b then there is a c such that precisely one of a ∧ c and b ∧ c is zero
worked very well. Thus defined concept (now called, in the dual form, the subfitness) turned out to be one of the most important weak separation properties suitable for the point-free context.
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Notes
- 1.
There are two equivalent constructions presented there, both of them, basically extending the reconstruction of a space X from Ω(X). In this section we have in mind the variant from A.3.4.2.
- 2.
- 3.
For example, one can find some interesting consequences in [204] where it appeared under the name of jointfit.
- 4.
T-uniformity in [203].
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Picado, J., Pultr, A. (2021). Subfitness and Basics of Fitness. In: Separation in Point-Free Topology. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-53479-0_2
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