Documentation

Mathlib.Topology.Algebra.Constructions.DomMulAct

Topological space structure on Mᵈᵐᵃ and Mᵈᵃᵃ #

In this file we define TopologicalSpace structure on Mᵈᵐᵃ and Mᵈᵃᵃ and prove basic theorems about these topologies. The topologies on Mᵈᵐᵃ and Mᵈᵃᵃ are the same as the topology on M. Formally, they are induced by DomMulAct.mk.symm and DomAddAct.mk.symm, since the types aren't definitionally equal.

Tags #

topological space, group action, domain action

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instance DomMulAct.instTopologicalSpace {M : Type u_1} [TopologicalSpace M] :
TopologicalSpace Mᵈᵐᵃ

Put the same topological space structure on Mᵈᵐᵃ as on the original space.

Equations
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instance DomAddAct.instTopologicalSpace {M : Type u_1} [TopologicalSpace M] :
TopologicalSpace Mᵈᵃᵃ

Put the same topological space structure on Mᵈᵃᵃ as on the original space.

Equations
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theorem DomMulAct.continuous_mk {M : Type u_1} [TopologicalSpace M] :
Continuous ⇑DomMulAct.mk
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theorem DomAddAct.continuous_mk {M : Type u_1} [TopologicalSpace M] :
Continuous ⇑DomAddAct.mk
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theorem DomMulAct.continuous_mk_symm {M : Type u_1} [TopologicalSpace M] :
Continuous ⇑DomMulAct.mk.symm
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theorem DomAddAct.continuous_mk_symm {M : Type u_1} [TopologicalSpace M] :
Continuous ⇑DomAddAct.mk.symm
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def DomMulAct.mkHomeomorph {M : Type u_1} [TopologicalSpace M] :
M ≃ₜ Mᵈᵐᵃ

DomMulAct.mk as a homeomorphism.

Equations
Instances For
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    def DomAddAct.mkHomeomorph {M : Type u_1} [TopologicalSpace M] :
    M ≃ₜ Mᵈᵃᵃ

    DomAddAct.mk as a homeomorphism.

    Equations
    Instances For
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      @[simp]
      theorem DomMulAct.mkHomeomorph_toEquiv {M : Type u_1} [TopologicalSpace M] :
      DomMulAct.mkHomeomorph.toEquiv = DomMulAct.mk
      source
      @[simp]
      theorem DomAddAct.mkHomeomorph_toEquiv {M : Type u_1} [TopologicalSpace M] :
      DomAddAct.mkHomeomorph.toEquiv = DomAddAct.mk
      source
      @[simp]
      theorem DomMulAct.coe_mkHomeomorph {M : Type u_1} [TopologicalSpace M] :
      ⇑DomMulAct.mkHomeomorph = ⇑DomMulAct.mk
      source
      @[simp]
      theorem DomAddAct.coe_mkHomeomorph {M : Type u_1} [TopologicalSpace M] :
      ⇑DomAddAct.mkHomeomorph = ⇑DomAddAct.mk
      source
      @[simp]
      theorem DomMulAct.coe_mkHomeomorph_symm {M : Type u_1} [TopologicalSpace M] :
      ⇑DomMulAct.mkHomeomorph.symm = ⇑DomMulAct.mk.symm
      source
      @[simp]
      theorem DomAddAct.coe_mkHomeomorph_symm {M : Type u_1} [TopologicalSpace M] :
      ⇑DomAddAct.mkHomeomorph.symm = ⇑DomAddAct.mk.symm
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      theorem DomMulAct.isInducing_mk {M : Type u_1} [TopologicalSpace M] :
      Topology.IsInducing ⇑DomMulAct.mk
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      theorem DomAddAct.isInducing_mk {M : Type u_1} [TopologicalSpace M] :
      Topology.IsInducing ⇑DomAddAct.mk
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      theorem DomMulAct.isEmbedding_mk {M : Type u_1} [TopologicalSpace M] :
      Topology.IsEmbedding ⇑DomMulAct.mk
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      theorem DomAddAct.isEmbedding_mk {M : Type u_1} [TopologicalSpace M] :
      Topology.IsEmbedding ⇑DomAddAct.mk
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      theorem DomMulAct.isOpenEmbedding_mk {M : Type u_1} [TopologicalSpace M] :
      Topology.IsOpenEmbedding ⇑DomMulAct.mk
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      theorem DomAddAct.isOpenEmbedding_mk {M : Type u_1} [TopologicalSpace M] :
      Topology.IsOpenEmbedding ⇑DomAddAct.mk
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      theorem DomMulAct.isClosedEmbedding_mk {M : Type u_1} [TopologicalSpace M] :
      Topology.IsClosedEmbedding ⇑DomMulAct.mk
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      theorem DomAddAct.isClosedEmbedding_mk {M : Type u_1} [TopologicalSpace M] :
      Topology.IsClosedEmbedding ⇑DomAddAct.mk
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      theorem DomMulAct.isQuotientMap_mk {M : Type u_1} [TopologicalSpace M] :
      Topology.IsQuotientMap ⇑DomMulAct.mk
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      theorem DomAddAct.isQuotientMap_mk {M : Type u_1} [TopologicalSpace M] :
      Topology.IsQuotientMap ⇑DomAddAct.mk
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      @[deprecated DomMulAct.isInducing_mk (since := "2024-10-28")]
      theorem DomMulAct.inducing_mk {M : Type u_1} [TopologicalSpace M] :
      Topology.IsInducing ⇑DomMulAct.mk

      Alias of DomMulAct.isInducing_mk.

      source
      @[deprecated DomMulAct.isEmbedding_mk (since := "2024-10-26")]
      theorem DomMulAct.embedding_mk {M : Type u_1} [TopologicalSpace M] :
      Topology.IsEmbedding ⇑DomMulAct.mk

      Alias of DomMulAct.isEmbedding_mk.

      source
      @[deprecated DomMulAct.isQuotientMap_mk (since := "2024-10-22")]
      theorem DomMulAct.quotientMap_mk {M : Type u_1} [TopologicalSpace M] :
      Topology.IsQuotientMap ⇑DomMulAct.mk

      Alias of DomMulAct.isQuotientMap_mk.

      source
      @[deprecated DomMulAct.isClosedEmbedding_mk (since := "2024-10-20")]
      theorem DomMulAct.closedEmbedding_mk {M : Type u_1} [TopologicalSpace M] :
      Topology.IsClosedEmbedding ⇑DomMulAct.mk

      Alias of DomMulAct.isClosedEmbedding_mk.

      source
      @[deprecated DomMulAct.isOpenEmbedding_mk (since := "2024-10-18")]
      theorem DomMulAct.openEmbedding_mk {M : Type u_1} [TopologicalSpace M] :
      Topology.IsOpenEmbedding ⇑DomMulAct.mk

      Alias of DomMulAct.isOpenEmbedding_mk.

      source
      theorem DomMulAct.isInducing_mk_symm {M : Type u_1} [TopologicalSpace M] :
      Topology.IsInducing ⇑DomMulAct.mk.symm
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      theorem DomAddAct.isInducing_mk_symm {M : Type u_1} [TopologicalSpace M] :
      Topology.IsInducing ⇑DomAddAct.mk.symm
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      theorem DomMulAct.isEmbedding_mk_symm {M : Type u_1} [TopologicalSpace M] :
      Topology.IsEmbedding ⇑DomMulAct.mk.symm
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      theorem DomAddAct.isEmbedding_mk_symm {M : Type u_1} [TopologicalSpace M] :
      Topology.IsEmbedding ⇑DomAddAct.mk.symm
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      @[deprecated DomMulAct.isInducing_mk_symm (since := "2024-10-28")]
      theorem DomMulAct.inducing_mk_symm {M : Type u_1} [TopologicalSpace M] :
      Topology.IsInducing ⇑DomMulAct.mk.symm

      Alias of DomMulAct.isInducing_mk_symm.

      source
      @[deprecated DomMulAct.isEmbedding_mk_symm (since := "2024-10-26")]
      theorem DomMulAct.embedding_mk_symm {M : Type u_1} [TopologicalSpace M] :
      Topology.IsEmbedding ⇑DomMulAct.mk.symm

      Alias of DomMulAct.isEmbedding_mk_symm.

      source
      theorem DomMulAct.isOpenEmbedding_mk_symm {M : Type u_1} [TopologicalSpace M] :
      Topology.IsOpenEmbedding ⇑DomMulAct.mk.symm
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      theorem DomAddAct.isOpenEmbedding_mk_symm {M : Type u_1} [TopologicalSpace M] :
      Topology.IsOpenEmbedding ⇑DomAddAct.mk.symm
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      theorem DomMulAct.isClosedEmbedding_mk_symm {M : Type u_1} [TopologicalSpace M] :
      Topology.IsClosedEmbedding ⇑DomMulAct.mk.symm
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      theorem DomAddAct.isClosedEmbedding_mk_symm {M : Type u_1} [TopologicalSpace M] :
      Topology.IsClosedEmbedding ⇑DomAddAct.mk.symm
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      @[deprecated DomMulAct.isClosedEmbedding_mk_symm (since := "2024-10-20")]
      theorem DomMulAct.closedEmbedding_mk_symm {M : Type u_1} [TopologicalSpace M] :
      Topology.IsClosedEmbedding ⇑DomMulAct.mk.symm

      Alias of DomMulAct.isClosedEmbedding_mk_symm.

      source
      theorem DomMulAct.isQuotientMap_mk_symm {M : Type u_1} [TopologicalSpace M] :
      Topology.IsQuotientMap ⇑DomMulAct.mk.symm
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      theorem DomAddAct.isQuotientMap_mk_symm {M : Type u_1} [TopologicalSpace M] :
      Topology.IsQuotientMap ⇑DomAddAct.mk.symm
      source
      @[deprecated DomMulAct.isQuotientMap_mk_symm (since := "2024-10-22")]
      theorem DomMulAct.quotientMap_mk_symm {M : Type u_1} [TopologicalSpace M] :
      Topology.IsQuotientMap ⇑DomMulAct.mk.symm

      Alias of DomMulAct.isQuotientMap_mk_symm.

      source
      instance DomMulAct.instT0Space {M : Type u_1} [TopologicalSpace M] [T0Space M] :
      T0Space Mᵈᵐᵃ
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      instance DomAddAct.instT0Space {M : Type u_1} [TopologicalSpace M] [T0Space M] :
      T0Space Mᵈᵃᵃ
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      instance DomMulAct.instT1Space {M : Type u_1} [TopologicalSpace M] [T1Space M] :
      T1Space Mᵈᵐᵃ
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      instance DomAddAct.instT1Space {M : Type u_1} [TopologicalSpace M] [T1Space M] :
      T1Space Mᵈᵃᵃ
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      instance DomMulAct.instT2Space {M : Type u_1} [TopologicalSpace M] [T2Space M] :
      T2Space Mᵈᵐᵃ
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      instance DomAddAct.instT2Space {M : Type u_1} [TopologicalSpace M] [T2Space M] :
      T2Space Mᵈᵃᵃ
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      instance DomMulAct.instT25Space {M : Type u_1} [TopologicalSpace M] [T25Space M] :
      T25Space Mᵈᵐᵃ
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      instance DomAddAct.instT25Space {M : Type u_1} [TopologicalSpace M] [T25Space M] :
      T25Space Mᵈᵃᵃ
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      instance DomMulAct.instT3Space {M : Type u_1} [TopologicalSpace M] [T3Space M] :
      T3Space Mᵈᵐᵃ
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      instance DomAddAct.instT3Space {M : Type u_1} [TopologicalSpace M] [T3Space M] :
      T3Space Mᵈᵃᵃ
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      instance DomMulAct.instT4Space {M : Type u_1} [TopologicalSpace M] [T4Space M] :
      T4Space Mᵈᵐᵃ
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      instance DomAddAct.instT4Space {M : Type u_1} [TopologicalSpace M] [T4Space M] :
      T4Space Mᵈᵃᵃ
      source
      instance DomMulAct.instT5Space {M : Type u_1} [TopologicalSpace M] [T5Space M] :
      T5Space Mᵈᵐᵃ
      source
      instance DomAddAct.instT5Space {M : Type u_1} [TopologicalSpace M] [T5Space M] :
      T5Space Mᵈᵃᵃ
      source
      instance DomMulAct.instR0Space {M : Type u_1} [TopologicalSpace M] [R0Space M] :
      R0Space Mᵈᵐᵃ
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      instance DomAddAct.instR0Space {M : Type u_1} [TopologicalSpace M] [R0Space M] :
      R0Space Mᵈᵃᵃ
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      instance DomMulAct.instR1Space {M : Type u_1} [TopologicalSpace M] [R1Space M] :
      R1Space Mᵈᵐᵃ
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      instance DomAddAct.instR1Space {M : Type u_1} [TopologicalSpace M] [R1Space M] :
      R1Space Mᵈᵃᵃ
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      instance DomMulAct.instRegularSpace {M : Type u_1} [TopologicalSpace M] [RegularSpace M] :
      RegularSpace Mᵈᵐᵃ
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      instance DomAddAct.instRegularSpace {M : Type u_1} [TopologicalSpace M] [RegularSpace M] :
      RegularSpace Mᵈᵃᵃ
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      instance DomMulAct.instNormalSpace {M : Type u_1} [TopologicalSpace M] [NormalSpace M] :
      NormalSpace Mᵈᵐᵃ
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      instance DomAddAct.instNormalSpace {M : Type u_1} [TopologicalSpace M] [NormalSpace M] :
      NormalSpace Mᵈᵃᵃ
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      instance DomMulAct.instCompletelyNormalSpace {M : Type u_1} [TopologicalSpace M] [CompletelyNormalSpace M] :
      CompletelyNormalSpace Mᵈᵐᵃ
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      instance DomAddAct.instCompletelyNormalSpace {M : Type u_1} [TopologicalSpace M] [CompletelyNormalSpace M] :
      CompletelyNormalSpace Mᵈᵃᵃ
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      instance DomMulAct.instDiscreteTopology {M : Type u_1} [TopologicalSpace M] [DiscreteTopology M] :
      DiscreteTopology Mᵈᵐᵃ
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      instance DomAddAct.instDiscreteTopology {M : Type u_1} [TopologicalSpace M] [DiscreteTopology M] :
      DiscreteTopology Mᵈᵃᵃ
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      instance DomMulAct.instSeparableSpace {M : Type u_1} [TopologicalSpace M] [TopologicalSpace.SeparableSpace M] :
      TopologicalSpace.SeparableSpace Mᵈᵐᵃ
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      instance DomAddAct.instSeparableSpace {M : Type u_1} [TopologicalSpace M] [TopologicalSpace.SeparableSpace M] :
      TopologicalSpace.SeparableSpace Mᵈᵃᵃ
      source
      instance DomMulAct.instFirstCountableTopology {M : Type u_1} [TopologicalSpace M] [FirstCountableTopology M] :
      FirstCountableTopology Mᵈᵐᵃ
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      instance DomAddAct.instFirstCountableTopology {M : Type u_1} [TopologicalSpace M] [FirstCountableTopology M] :
      FirstCountableTopology Mᵈᵃᵃ
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      instance DomMulAct.instSecondCountableTopology {M : Type u_1} [TopologicalSpace M] [SecondCountableTopology M] :
      SecondCountableTopology Mᵈᵐᵃ
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      instance DomAddAct.instSecondCountableTopology {M : Type u_1} [TopologicalSpace M] [SecondCountableTopology M] :
      SecondCountableTopology Mᵈᵃᵃ
      source
      instance DomMulAct.instCompactSpace {M : Type u_1} [TopologicalSpace M] [CompactSpace M] :
      CompactSpace Mᵈᵐᵃ
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      instance DomAddAct.instCompactSpace {M : Type u_1} [TopologicalSpace M] [CompactSpace M] :
      CompactSpace Mᵈᵃᵃ
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      instance DomMulAct.instLocallyCompactSpace {M : Type u_1} [TopologicalSpace M] [LocallyCompactSpace M] :
      LocallyCompactSpace Mᵈᵐᵃ
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      instance DomAddAct.instLocallyCompactSpace {M : Type u_1} [TopologicalSpace M] [LocallyCompactSpace M] :
      LocallyCompactSpace Mᵈᵃᵃ
      source
      instance DomMulAct.instWeaklyLocallyCompactSpace {M : Type u_1} [TopologicalSpace M] [WeaklyLocallyCompactSpace M] :
      WeaklyLocallyCompactSpace Mᵈᵐᵃ
      source
      instance DomAddAct.instWeaklyLocallyCompactSpace {M : Type u_1} [TopologicalSpace M] [WeaklyLocallyCompactSpace M] :
      WeaklyLocallyCompactSpace Mᵈᵃᵃ
      source
      @[simp]
      theorem DomMulAct.map_mk_nhds {M : Type u_1} [TopologicalSpace M] (x : M) :
      Filter.map (⇑DomMulAct.mk) (nhds x) = nhds (DomMulAct.mk x)
      source
      @[simp]
      theorem DomAddAct.map_mk_nhds {M : Type u_1} [TopologicalSpace M] (x : M) :
      Filter.map (⇑DomAddAct.mk) (nhds x) = nhds (DomAddAct.mk x)
      source
      @[simp]
      theorem DomMulAct.map_mk_symm_nhds {M : Type u_1} [TopologicalSpace M] (x : Mᵈᵐᵃ) :
      Filter.map (⇑DomMulAct.mk.symm) (nhds x) = nhds (DomMulAct.mk.symm x)
      source
      @[simp]
      theorem DomAddAct.map_mk_symm_nhds {M : Type u_1} [TopologicalSpace M] (x : Mᵈᵃᵃ) :
      Filter.map (⇑DomAddAct.mk.symm) (nhds x) = nhds (DomAddAct.mk.symm x)
      source
      @[simp]
      theorem DomMulAct.comap_mk_nhds {M : Type u_1} [TopologicalSpace M] (x : Mᵈᵐᵃ) :
      Filter.comap (⇑DomMulAct.mk) (nhds x) = nhds (DomMulAct.mk.symm x)
      source
      @[simp]
      theorem DomAddAct.comap_mk_nhds {M : Type u_1} [TopologicalSpace M] (x : Mᵈᵃᵃ) :
      Filter.comap (⇑DomAddAct.mk) (nhds x) = nhds (DomAddAct.mk.symm x)
      source
      @[simp]
      theorem DomMulAct.comap_mk.symm_nhds {M : Type u_1} [TopologicalSpace M] (x : M) :
      Filter.comap (⇑DomMulAct.mk.symm) (nhds x) = nhds (DomMulAct.mk x)
      source
      @[simp]
      theorem DomAddAct.comap_mk.symm_nhds {M : Type u_1} [TopologicalSpace M] (x : M) :
      Filter.comap (⇑DomAddAct.mk.symm) (nhds x) = nhds (DomAddAct.mk x)