235893.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 10 curated AI-written annotations and 7190 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

9 declarations, 2 contain sorry, 0 declared as axioms

Mathoverflow 235893

Reference: mathoverflow/235893 asked by user Willie Wong

Imports (1)

FormalConjecturesUtil

def Mathoverflow235893.IsConnectedMap For topological spaces $X$ and $Y$ we say a function $f : X → Y$ is connected is it sends connected sets to connected sets line 38 source
Source documentation
For topological spaces $X$ and $Y$ we say a function $f : X → Y$ is connected is it sends connected sets to connected sets.

no axioms what this means
Statement uses: TopologicalSpace
Proof uses: Set, IsConnected, Set.image
```
def IsConnectedMap {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] (f : X → Y) : Prop :=
…
```
theorem Mathoverflow235893.Continuous.isConnectedMap By a standard result, every continuous map is connected line 45 source
Source documentation
By a standard result, every continuous map is connected

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: TopologicalSpace, Continuous, Mathoverflow235893.IsConnectedMap
Proof uses: Set, IsConnected, IsConnected.image, Continuous.continuousOn
```
theorem Continuous.isConnectedMap {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]
    {f : X → Y} (hf : Continuous f) : IsConnectedMap f :=
…
```
Mentions: Mathoverflow235893.IsConnectedMap
lemma Mathoverflow235893.isConnected_iff_ordConnected_and_nonempty A set in $\mathbb{R}$ is connected if and only if it is order-connected and non-empty line 53 source
Source documentation
A set in $\mathbb{R}$ is connected if and only if it is order-connected and non-empty.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Real, Iff, IsConnected, And, Set.OrdConnected, Set.Nonempty
Proof uses: Iff.intro, IsPreconnected, And.intro, IsPreconnected.ordConnected, Set.OrdConnected.isPreconnected
29 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
```
lemma isConnected_iff_ordConnected_and_nonempty {s : Set ℝ} :
    IsConnected s ↔ s.OrdConnected ∧ s.Nonempty := by
…
```
lemma Mathoverflow235893.isConnectedMap_symm_of_R If $f : \mathbb{R} \to \mathbb{R}$ is a connected bijection, then its inverse is also a connected bijection line 67 source
Source documentation
If $f : \mathbb{R} \to \mathbb{R}$ is a connected bijection, then its inverse is also a connected bijection.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Equiv, Real, Mathoverflow235893.IsConnectedMap, Equiv.symm
Proof uses: Set, IsConnected, Eq.mpr, Set.image, And, Set.OrdConnected, Set.Nonempty, id, Eq, congrArg, propext, Mathoverflow235893.isConnected_iff_ordConnected_and_nonempty, Eq.mp, Membership.mem, Exists.intro, Set.mem_image_of_mem, And.intro, LE.le, Set.Icc, Set.ordConnected_def and 93 more
81 instances, coercions and auxiliary constants left out.

Tactic trace: 84 steps
Open to load the checked before-and-after goal states.
```
lemma isConnectedMap_symm_of_R (f : ℝ ≃ ℝ) (hf : IsConnectedMap f) : IsConnectedMap f.symm := by
…
```
Mentions: Mathoverflow235893.IsConnectedMap
lemma Mathoverflow235893.isConnectedMap_comp The composition of two connected maps is a connected map line 153 source
Source documentation
The composition of two connected maps is a connected map.

standard axioms: Quot.sound, propext what this means
Statement uses: TopologicalSpace, Mathoverflow235893.IsConnectedMap, Function.comp
Proof uses: Set, IsConnected, Eq.mpr, Set.image, id, Eq, congrArg, Set.image_comp

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
lemma isConnectedMap_comp {X Y Z : Type*} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z]
    {f : X → Y} {g : Y → Z} (hf : IsConnectedMap f) (hg : IsConnectedMap g) :
    IsConnectedMap (g ∘ f) := by
…
```
Mentions: Mathoverflow235893.IsConnectedMap
lemma Mathoverflow235893.isConnectedMap_homeomorph A homeomorphism is a connected map line 167 source
Source documentation
A homeomorphism is a connected map.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: TopologicalSpace, Homeomorph, Mathoverflow235893.IsConnectedMap
Proof uses: Set, IsConnected, IsConnected.image, Continuous.continuousOn, Homeomorph.continuous
6 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
lemma isConnectedMap_homeomorph {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]
    (h : X ≃ₜ Y) : IsConnectedMap h := by
…
```
Mentions: Mathoverflow235893.IsConnectedMap
lemma Mathoverflow235893.isConnectedMap_symm_of_E1 If $f : \mathbb{R}^1 \to \mathbb{R}^1$ is a connected bijection, then its inverse is also a connected bijection line 179 source
Source documentation
If $f : \mathbb{R}^1 \to \mathbb{R}^1$ is a connected bijection, then its inverse is also a connected bijection.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Equiv, EuclideanSpace, Real, Fin, OfNat.ofNat, Nat, Mathoverflow235893.IsConnectedMap, ENNReal, Equiv.symm
Proof uses: Homeomorph, Homeomorph.trans, ContinuousLinearEquiv.toHomeomorph, RingHom.id, EuclideanSpace.equiv, Homeomorph.funUnique, Equiv.trans, Homeomorph.toEquiv, Homeomorph.symm, Mathoverflow235893.isConnectedMap_homeomorph, id, Mathoverflow235893.isConnectedMap_comp, Function.comp, Mathoverflow235893.isConnectedMap_symm_of_R, Eq, funext, PiLp.ext, of_eq_true, WithLp.ofLp, Eq.trans and 6 more
54 instances, coercions and auxiliary constants left out.

Tactic trace: 16 steps
Open to load the checked before-and-after goal states.
```
lemma isConnectedMap_symm_of_E1 (f : ℝ^1 ≃ ℝ^1) (hf : IsConnectedMap f) : IsConnectedMap f.symm := by
…
```
Mentions: Mathoverflow235893.IsConnectedMap
theorem Mathoverflow235893.mathoverflow_235893 Assume for $n>1$, $f:\mathbb{R}^n\to\mathbb{R}^n$ is a bijection, where $\mathbb{R}^n$ is equipped with the standard topology sorry line 207 source
Source documentation
Assume for $n>1$, $f:\mathbb{R}^n\to\mathbb{R}^n$ is a bijection, where $\mathbb{R}^n$ is equipped with the standard topology. Does the connectedness of (the induced power set map) $f$ imply that of $f^{-1}$?

depends on sorry what this means
Statement uses: Iff, True, Nat, GT.gt, OfNat.ofNat, Equiv, EuclideanSpace, Real, Fin, Mathoverflow235893.IsConnectedMap, ENNReal, Equiv.symm
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem mathoverflow_235893 :
    answer(sorry) ↔ ∀ n > 1, ∀ (f : ℝ^n ≃ ℝ^n), IsConnectedMap f → IsConnectedMap f.symm := by
…
```
Mentions: Mathoverflow235893.IsConnectedMap
theorem Mathoverflow235893.mathoverflow_260589 There exists a connected bijection ℝ → ℝ^2 where the inverse is not connected, proven in mathoverflow/260589 by user Gro-Tsen sorry line 217 source
Source documentation
There exists a connected bijection ℝ → ℝ^2 where the inverse is not connected, proven in mathoverflow/260589 by user Gro-Tsen.

L4DEL explanation
There exists a connected bijection ℝ → ℝ^2 where the inverse is not connected, proven in mathoverflow/260589 by user Gro-Tsen.
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Exists, Equiv, Real, EuclideanSpace, Fin, OfNat.ofNat, Nat, And, Mathoverflow235893.IsConnectedMap, ENNReal, Not, Equiv.symm
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem mathoverflow_260589 :
    ∃ f : ℝ ≃ ℝ^2, IsConnectedMap f ∧ ¬ IsConnectedMap f.symm := by
…
```
Mentions: Mathoverflow235893.IsConnectedMap