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 $(X,\mathcal{ J })$ 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:

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

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