I find the one on wikipedia to be a bit intimidating. It says that...(adsbygoogle = window.adsbygoogle || []).push({});

"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."

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# More docile dfn of local path-connectedness?

**Physics Forums | Science Articles, Homework Help, Discussion**