# Locally path connected

1. Jul 4, 2013

### hedipaldi

Hi,
I am trying to prove that any compact metric space that is also locally connected,must be locally path connected.
can someone help?

2. Jul 4, 2013

### micromass

Do you know that a locally connected and connected metric space is locally path connected?

To prove that, consider an element x the set of all y such that there is a path from x to y.

3. Jul 4, 2013

### hedipaldi

You mean to show that this is clopen?This is my difficulty.I suppose the compacity is needed

Last edited: Jul 4, 2013
4. Jul 5, 2013

### micromass

Check Willard theorem 31.2.

5. Jul 6, 2013

### hedipaldi

hahn mazurkiewicz theorem gives the answer