Theorems · Theorem · general topology

locPathConnectedSpace_iff_isOpen_pathComponentIn

Deprecated since 2026-06-21Use locallyPathConnectedSpace_iff_isOpen_pathComponentIn instead.

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

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

Defined in

Mathlib.Topology.Connected.LocallyPathConnected

Cited by

0 results in Mathlib

Foundations

Depth 131 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpace