A topological space \( X \) is called to be locally path-connected if it admits a basis of path-connected open subsets.
Remarks
- Every locally path-connected space is also locally connected, since path-connected subsets are connected .
- Every path-component of \(X\) is open.
- Components of \(X\) correspond with its path components.
- \(X\) is connected if and only if it is path-connected.