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Mathlib.Topology.Instances.Sign

Topology on SignType #

This file gives SignType the discrete topology, and proves continuity results for SignType.sign in an OrderTopology.

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instance instTopologicalSpaceSignType :
TopologicalSpace SignType
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    instance instDiscreteTopologySignType :
    DiscreteTopology SignType
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    theorem continuousAt_sign_of_pos {α : Type u_1} [Zero α] [TopologicalSpace α] [PartialOrder α] [DecidableLT α] [OrderTopology α] {a : α} (h : 0 < a) :
    ContinuousAt (⇑SignType.sign) a
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    theorem continuousAt_sign_of_neg {α : Type u_1} [Zero α] [TopologicalSpace α] [PartialOrder α] [DecidableLT α] [OrderTopology α] {a : α} (h : a < 0) :
    ContinuousAt (⇑SignType.sign) a
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    theorem continuousAt_sign_of_ne_zero {α : Type u_1} [Zero α] [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α} (h : a ≠ 0) :
    ContinuousAt (⇑SignType.sign) a