# Thread: Local Compactness

1. ## Local Compactness

Dear Colleagues,

A metric space $X$ is said to be locally compact if every point of $X$ has a compact neighborhood. Show that a compact metric space $X$ is locally compact.

Regards,

Raed.

2. Originally Posted by raed
Dear Colleagues,

A metric space $X$ is said to be locally compact if every point of $X$ has a compact neighborhood. Show that a compact metric space $X$ is locally compact.

Regards,

Raed.

X is a neighborhood of each and every point in it, so...

Tonio

3. For every $p\in X$ , $X$ is a compact neighbourhood of $p$ .

Edited: Sorry, I didn't see the previous answer.

4. Thank you very much.