Loogle!

Result

Found 27 declarations mentioning ConnectedSpace. Of these, 11 match your pattern(s).

• IrreducibleSpace.connectedSpace 📋 Mathlib.Topology.Connected.Basic
  (α : Type u) [TopologicalSpace α] [IrreducibleSpace α] : ConnectedSpace α
• ConnectedSpace.mk 📋 Mathlib.Topology.Connected.Basic
  {α : Type u} [TopologicalSpace α] [toPreconnectedSpace : PreconnectedSpace α] (toNonempty : Nonempty α) : ConnectedSpace α
• Quotient.instConnectedSpace 📋 Mathlib.Topology.Connected.Basic
  {α : Type u} [TopologicalSpace α] {s : Setoid α} [ConnectedSpace α] : ConnectedSpace (Quotient s)
• instConnectedSpaceForall 📋 Mathlib.Topology.Connected.Basic
  {ι : Type u_1} {π : ι → Type u_2} [(i : ι) → TopologicalSpace (π i)] [∀ (i : ι), ConnectedSpace (π i)] : ConnectedSpace ((i : ι) → π i)
• instConnectedSpaceProd 📋 Mathlib.Topology.Connected.Basic
  {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] [ConnectedSpace α] [ConnectedSpace β] : ConnectedSpace (α × β)
• Function.Surjective.connectedSpace 📋 Mathlib.Topology.Connected.Basic
  {α : Type u} {β : Type v} [TopologicalSpace α] [ConnectedSpace α] [TopologicalSpace β] {f : α → β} (hf : Function.Surjective f) (hf' : Continuous f) : ConnectedSpace β
• Subtype.connectedSpace 📋 Mathlib.Topology.Connected.Basic
  {α : Type u} [TopologicalSpace α] {s : Set α} (h : IsConnected s) : ConnectedSpace ↑s
• AddTorsor.connectedSpace 📋 Mathlib.Topology.Algebra.MulAction
  (G : Type u_1) (P : Type u_2) [AddGroup G] [AddTorsor G P] [TopologicalSpace G] [PreconnectedSpace G] [TopologicalSpace P] [ContinuousVAdd G P] : ConnectedSpace P
• unitInterval.instConnectedSpaceElemReal 📋 Mathlib.Topology.UnitInterval
  : ConnectedSpace ↑unitInterval
• PathConnectedSpace.connectedSpace 📋 Mathlib.Topology.Connected.PathConnected
  {X : Type u_1} [TopologicalSpace X] [PathConnectedSpace X] : ConnectedSpace X
• OnePoint.instConnectedSpaceOfPreconnectedSpaceOfNoncompactSpace 📋 Mathlib.Topology.Compactification.OnePoint
  {X : Type u_1} [TopologicalSpace X] [PreconnectedSpace X] [NoncompactSpace X] : ConnectedSpace (OnePoint X)

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using (by default) Ctrl-K Ctrl-S. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

1. By constant:
   🔍 Real.sin
   finds all lemmas whose statement somehow mentions the sine function.

2. By lemma name substring:
   🔍 "differ"
   finds all lemmas that have "differ" somewhere in their lemma name.

3. By subexpression:
   🔍 _ * (_ ^ _)
   finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

   The pattern can also be non-linear, as in
   🔍 Real.sqrt ?a * Real.sqrt ?a

   If the pattern has parameters, they are matched in any order. Both of these will find List.map:
   🔍 (?a -> ?b) -> List ?a -> List ?b
   🔍 List ?a -> (?a -> ?b) -> List ?b

4. By main conclusion:
   🔍 |- tsum _ = _ * tsum _
   finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

   As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
   🔍 |- _ < _ → tsum _ < tsum _
   will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO.

This is Loogle revision 19971e9 serving mathlib revision 4cb993b