Toronto spaces
Toronto spaces
Problem statement
DiscreteTopology.of_t2_of_torontoSpace
Any T2, Toronto space is discrete.
Lean 4
Formal statement
Lean 4/-
Copyright 2026 The Formal Conjectures Authors.
Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at
https://www.apache.org/licenses/LICENSE-2.0
Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
-/
import FormalConjecturesUtil
/-!
# 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](https://en.wikipedia.org/wiki/Toronto_space)
- [The Toronto problem](https://wrbrian.wordpress.com/wp-content/uploads/2012/01/thetorontoproblem.pdf) by *W.R. Brian*
-/
namespace Toronto
open Cardinal
variable (X : Type*) [TopologicalSpace X]
/-- A *Toronto space* is a topological space
which is homeomorphic to all of its subspaces of same cardinality.
-/
class TorontoSpace where
toronto : ∀ ⦃Y : Set X⦄, #Y = #X → Y ≃ₜ X
/-- Every finite space is Toronto, since
the only subspace with same cardinality is the space itself.
-/
@[category test, AMS 54]
instance Finite.torontoSpace [Finite X] : TorontoSpace X where
toronto := by
intro Y hY
have eq : Y = Set.univ := by
refine (Set.eq_univ_iff_ncard Y).mpr ?_
have : Y.ncard = (#↑Y).toNat := by
exact rfl
rw [this, hY]
exact rfl
rw [eq]
exact Homeomorph.Set.univ X
/-- Any T2, Toronto space is discrete. -/
@[category research open, AMS 54]
theorem DiscreteTopology.of_t2_of_torontoSpace [T2Space X] [TorontoSpace X] :
DiscreteTopology X := by
sorry
-- TODO: Add implication that this holds under the GCH
end Toronto
On-site verification needs an executable Lean target. None is attached yet.
A source attachment or link is not a verified proof. The statement’s correspondence with the original question needs human review.
Proofs & verification
No Lean has been submitted for this problem yet. A Lean file can be attached to a submission for review, though with no target here it cannot be machine-checked.
Background
Source snapshot inspected on 10 September 2026. Subsequent research status and the accuracy of source formalizations have not been independently certified.
Fixed commit b82b08faa9006484021c12005ab41287fb2ffb69. This file contains 1 source-labeled research-open declarations. Other declarations in the attached file can be definitions, solved results or tests. The file is preserved as source material; it has not been compiled or proved by this import.
Copyright The Formal Conjectures Authors. Code: Apache-2.0; other original material: CC BY 4.0. Third-party sources retain their own licenses. Original source files and notices preserved.
Recorded progress
Add yours →Nothing recorded yet. A pointer to a paper the entry is missing counts.
Machine attempts
0 recordedAttempts run against the statement below and are recorded in full, including the failures. The point is to stop the next model — and the next person — from re-walking a road that is already known to end. Each attempt costs 25 credits.
No machine attempts recorded.
Sign in to run an attempt. New accounts start with enough credits for ten.
Discussion
0 notesNotes are for clarifying the statement, flagging a missing reference, or recording why an approach fails. Claimed solutions belong in a submission, where they can be reviewed.
No notes yet.
Sign in to add a note.