locallyPathConnectedSpace_iff_isOpen_pathComponentIn

∀ {X : Type u_4} [inst : TopologicalSpace X],
  LocallyPathConnectedSpace X ↔ ∀ (x : X) (u : Set X), IsOpen u → IsOpen (pathComponentIn u x)

A space is locally path-connected iff all path components of open subsets are open.

Defined in: Mathlib.Topology.Connected.LocallyPathConnected
Cited by: 3 results in Mathlib
Foundations: Depth 130 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: TopologicalSpace

Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Results whose statement or proof uses this declaration.