CompactOpen.lean`

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Joint continuity of pairing into a product

This file shows that the pairing C(α, β) × C(α, δ) → C(α, β × δ) is continuous in the compact-open topologies whenever α is an R₁ space, with no local compactness assumption.

Material destined for Mathlib.Topology.CompactOpen.

Imports (1)

Mathlib.Topology.CompactOpen

Imported by 2 files

lemma ContinuousMap.continuous_prodMk Documented declaration: continuous_prodMk line 27 source
Source documentation
The pairing map C(α, β) × C(α, δ) → C(α, β × δ) is continuous in the compact-open topologies when α is an R₁ space, for instance a topological group, with no assumptions on β and δ. No local compactness of α is required: a compact set mapped into an open set of β × δ splits into finitely many compact pieces on each of which both components map into an open rectangle.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: TopologicalSpace, R1Space, Continuous, Prod, ContinuousMap, ContinuousMap.prodMk, Prod.fst, Prod.snd
Proof uses: Eq.mpr, ContinuousAt, id, Eq, Filter.Tendsto, nhds, Set, IsCompact, IsOpen, Set.MapsTo, Filter.Eventually, forall_congr, Prod.mk, Membership.mem, Exists, And, LE.le, SProd.sprod, Iff.mp, isOpen_prod_iff and 39 more
21 instances, coercions and auxiliary constants left out.

Tactic trace: 17 steps
Open to load the checked before-and-after goal states.
```
lemma ContinuousMap.continuous_prodMk {α β δ : Type*} [TopologicalSpace α] [R1Space α]
    [TopologicalSpace β] [TopologicalSpace δ] :
    Continuous fun p : C(α, β) × C(α, δ) ↦ p.1.prodMk p.2 := by
…
```