A question about compact and complete metric spaces

Question

I have a question about compact and complete metric space.

These two concepts how related to each other.

Is compact metric space complete?

If the question is elementary I apologize you.

Thank you.

score 2 · Accepted Answer · answered Mar 09 '15 at 12:48

2

For metric spaces, compact means "every sequence has a convergent subsequence". Complete means "every Cauchy sequence is convergent".

For any metric space, a Cauchy sequence is convergent if it has a convergent subsequence.

So, if a space is compact, then every Cauchy sequence has a convergent subsequence, and is thus itself convergent. So, every compact space is complete.

answered Mar 09 '15 at 12:48

Ben Grossmann

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OK. Thank you so much. – utya Mar 09 '15 at 12:49
1

@utya please accept the answer if you found the answer helpful (by pressing the check mark). :) – Pedro Mar 09 '15 at 13:00
But (on the other side) not every complete metric space is compact. – GEdgar Mar 09 '15 at 13:49

A question about compact and complete metric spaces

1 Answers1