Theorems · Theorem · order theory

Topology.IsLowerSet.monotone_iff_continuous

∀ {α : Type u_1} {β : Type u_2} [inst : Preorder α] [inst_1 : Preorder β] [inst_2 : TopologicalSpace α]
  [inst_3 : TopologicalSpace β] [Topology.IsLowerSet α] [Topology.IsLowerSet β] {f : α → β}, Monotone f ↔ Continuous f

Defined in: Mathlib.Topology.Order.UpperLowerSetTopology
Cited by: 0 results in Mathlib
Foundations: Depth 69 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: Preorder Preorder TopologicalSpace TopologicalSpace Topology.IsLowerSet Topology.IsLowerSet

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TopologicalSpacestatement and proof · cited by 24,529
Preorderstatement and proof · cited by 7,952
Continuousstatement and proof · cited by 2,592
Monotonestatement · cited by 1,397
Topology.IsLowerSetstatement and proof · cited by 17
monotone_dual_iffproof · cited by 2
Topology.IsUpperSet.monotone_iff_continuousproof · cited by 1

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