Theorems · Theorem · general topology

Filter.map_atTop_eq_of_gc_preorder

∀ {α : Type u_3} {β : Type u_4} [inst : Preorder α] [IsDirectedOrder α] [inst_2 : Preorder β] [IsDirectedOrder β]
  {f : α → β},
  Monotone f →
    ∀ (b : β), (∀ (c : β), b ≤ c → ∃ x, f x = c ∧ ∀ (a : α), f a ≤ c ↔ a ≤ x) → Filter.map f Filter.atTop = Filter.atTop

A function f maps upwards closed sets (atTop sets) to upwards closed sets when it is a Galois insertion. The Galois "insertion" and "connection" is weakened to only require it to be an insertion and a connection above b.

Defined in

Mathlib.Order.Filter.AtTopBot.Basic

Cited by

3 results in Mathlib

Foundations

Depth 65 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

PreorderIsDirectedOrderPreorderIsDirectedOrder