Locally path-connected topological space

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Not to be confused with: locally connected topological space - note that locally path-connected is to path connected as locally connected is to connected

Definition

Let [ilmath](X,\mathcal{ J })[/ilmath] be a topological space, we say that it is locally path-connected or is a locally path-connected (topological space) (Caveat:path-connected is a related but distinct concept) if it satisfies the following property:

• [ilmath]\exists\mathcal{B}\in\mathcal{P}(\mathcal{P}(X))\forall B\in\mathcal{B}[B\text{ is } [/ilmath][ilmath]\text{path-connected} [/ilmath][ilmath]][/ilmath] -
  TODO: Presumably
  [ilmath]B[/ilmath] is considered as a subspace of [ilmath]X[/ilmath].
  • In words: [ilmath](X,\mathcal{ J })[/ilmath] is locally path-connected if it admits a basis of path-connected (open by definition) subsets of [ilmath]X[/ilmath].

See next

• Locally path-connected implies locally connected
• Every open set of a locally path-connected space is locally path-connected
• If a space is locally path-connected then it connected if and only if it is path-connected
• If a space is locally path-connected then the path-connected-components of the space are equal to its connected-components

See also

• Path-connected
• Connected
  • Locally connected - note that locally path-connected is to path connected as locally connected is to connected