Prove Compact Metric Space is Locally Path Connected

[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?
thank's in advance.

 

[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.

 

[hedipaldi]

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

 

[micromass]

Check Willard theorem 31.2.

 

[hedipaldi]

hahn mazurkiewicz theorem gives the answer