Topology on Hom(R, S)
#
In this file, we define topology on Hom(A, R)
for a topological ring R
,
given by the coarsest topology that makes f ↦ f x
continuous for all x : A
.
Alternatively, given a presentation A = ℤ[xᵢ]/I
,
this is the subspace topology Hom(A, R) ↪ Hom(ℤ[xᵢ], R) = Rᶥ
.
Main results #
CommRingCat.HomTopology.isClosedEmbedding_precomp_of_surjective
:Hom(A/I, R)
is a closed subspace ofHom(A, R)
ifR
is Hausdorff.CommRingCat.HomTopology.mvPolynomialHomeo
:Hom(A[Xᵢ], R)
is homeomorphic toHom(A, R) × Rᶥ
.CommRingCat.HomTopology.isEmbedding_pushout
:Hom(B ⊗[A] C, R)
has the subspace topology fromHom(B, R) × Hom(C, R)
.
The topology on Hom(A, R)
for a topological ring R
, given by the coarsest topology that
makes f ↦ f x
continuous for all x : A
(see continuous_apply
).
Alternatively, given a presentation A = ℤ[xᵢ]/I
,
This is the subspace topology Hom(A, R) ↪ Hom(ℤ[xᵢ], R) = Rᶥ
(see mvPolynomialHomeo
).
This is a scoped instance in CommRingCat.HomTopology
.
Equations
- CommRingCat.HomTopology.instTopologicalSpaceHom = TopologicalSpace.induced (fun (f : A ⟶ R) => ⇑(CommRingCat.Hom.hom f)) inferInstance
Instances For
If A ≅ B
, then Hom(A, R)
is homeomorphc to Hom(B, R)
.
Equations
- One or more equations did not get rendered due to their size.
Instances For
Hom(A/I, R)
has the subspace topology of Hom(A, R)
.
More generally, a surjection A ⟶ B
gives rise to an embedding Hom(B, R) ⟶ Hom(A, R)
Hom(A/I, R)
is a closed subspace of Hom(A, R)
if R
is T1.
Hom(A[Xᵢ], R)
is homeomorphic to Hom(A, R) × Rⁱ
.
Equations
- One or more equations did not get rendered due to their size.
Instances For
Hom(B ⊗[A] C, R)
has the subspace topology from Hom(B, R) × Hom(C, R)
.