0
votes
0answers
55 views

The concept of compactness

What is the meaning of, " a metric space is compact" ? I know the formal definition but i don't really understand it's true meaning, what is the concept behind compactness ? I seems like a lot of ...
63
votes
12answers
5k views

Why is compactness so important?

I've read many times that 'compactness' is such an extremely important and useful concept, though it's still not very apparent why. The only theorems I've seen concerning it are the Heine-Borel ...
3
votes
1answer
149 views

Intuition behind compact subspaces of a metric space

I've read up on compactness in a metric space and have found a few definitions (let $X$ be a metric space and $E \subset X$ in all the following): $E$ is compact in $X$ if for every open covering of ...
14
votes
8answers
2k views

How to understand compactness? [duplicate]

How to understand the compactness in topology space in intuitive way?