There are many nice discussions here (see disussion1,discussion2,discussion3) to show compactness is kind of 'finiteness', which means some properties of finite set can then be used for infinite set, in the way of local-to-global principles. Or, as said "finiteness = compactness + discreteness".
My question is: why we use word "compact" to describe this property? By literal meaning, compact means tightly packed. So, is there anything really compact when we have the property of compactness ?
Yes. In a compact space, every infinite set has a limit point. Therefore, infinite subsets of compact spaces cannot be spread, in the sense that each point is far from the other ones.