The Frame of Nuclei on an Alexandroff Space


Let \(\mathcal {O} S\) be the frame of open sets of a topological space S, and let \(N(\mathcal {O} S)\) be the frame of nuclei on \(\mathcal {O} S\). For an Alexandroff space S, we prove that \(N(\mathcal {O} S)\) is spatial iff the infinite binary tree \({\mathscr{T}}_2\) does not embed isomorphically into (S,≤), where ≤ is the specialization preorder of S.

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Ávila, F., Bezhanishvili, G., Morandi, P.J. et al. The Frame of Nuclei on an Alexandroff Space. Order 38, 67–78 (2021).

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  • Frame
  • Locale
  • Nucleus
  • Priestley space
  • Alexandroff space
  • Partial order
  • Total order
  • Tree