homeoWithUpperSetTopologyorderIso

English

An ordered set is isomorphic to the upper set topology on its specialization order via a canonical homeomorphism.

Русский

Корректное домашнее отображение устанавливает гомоморфизм между порядком и верхним множеством специализации.

LaTeX

$$$\text{homeoWithUpperSetTopologyorderIso}(\alpha):\alpha \cong_t \text{WithUpperSet}(\text{Specialization}(\alpha))$$$

Lean4

/-- An Alexandrov-discrete space is isomorphic to the upper set topology of its specialisation
order. -/
def homeoWithUpperSetTopologyorderIso (α : Type*) [TopologicalSpace α] [AlexandrovDiscrete α] :
    α ≃ₜ WithUpperSet (Specialization α) :=
  (toEquiv.trans toUpperSet).toHomeomorph fun s ↦ by simp [Set.preimage_comp]

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