Toronto.lean`

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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

3 declarations, 1 contain sorry, 0 declared as axioms

Toronto spaces

A Toronto space is a topological space which is homeomorphic to all of its subspaces of same cardinality.

It is conjectured that every T2, Toronto space is discrete. W.R. Brian proved that this holds under GCH.

References:

Wikipedia
The Toronto problem by W.R. Brian

Imports (1)

FormalConjecturesUtil

class Toronto.TorontoSpace A Toronto space is a topological space which is homeomorphic to all of its subspaces of same cardinality line 45 source
Source documentation
A Toronto space is a topological space which is homeomorphic to all of its subspaces of same cardinality.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: TopologicalSpace
```
class TorontoSpace where
…
```
instance Toronto.Finite.torontoSpace Every finite space is Toronto, since the only subspace with same cardinality is the space itself line 52 source
Source documentation
Every finite space is Toronto, since the only subspace with same cardinality is the space itself.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: TopologicalSpace, Finite, Toronto.TorontoSpace
Proof uses: Toronto.TorontoSpace.mk, Set, Eq, Cardinal, Cardinal.mk, Set.univ, Eq.mpr, Homeomorph, Membership.mem, Homeomorph.Set.univ
5 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
```
instance Finite.torontoSpace [Finite X] : TorontoSpace X where
…
```
Mentions: Toronto.TorontoSpace
theorem Toronto.DiscreteTopology.of_t2_of_torontoSpace Any T2, Toronto space is discrete sorry line 66 source
Source documentation
Any T2, Toronto space is discrete.

depends on sorry what this means
Statement uses: TopologicalSpace, T2Space, Toronto.TorontoSpace, DiscreteTopology
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem DiscreteTopology.of_t2_of_torontoSpace [T2Space X] [TorontoSpace X] :
    DiscreteTopology X := by
…
```
Mentions: Toronto.TorontoSpace