I find the one on wikipedia to be a bit intimidating. It says that... "A topological space is said to be locally path-connected if it has a base of path-connected sets." We have really not covered this notion of "base" in my course, and even though I know what it is, I don't have any experience working with it. For instance, my anxieties stem from... can every topology be generated by a basis? If a topology can be generated by a basis, is it unique? If no, can a topology admit a basis of path-connected elements as well as a basis of non-path connected elements? So is there a friendly caracterisation of this notion? A natural candidate that immidiately comes to mind would be "A topological space X is said to be locally path-connected if every nbh of every point is itself a path-connected subspace of X."