5

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 ?

Asaf Karagila
  • 393,674
X Leo
  • 508
  • 2
    Well, a compact disc is certainly compact. Maybe not as much as a minidisc, but certainly compared to a vinyl record. – Asaf Karagila Aug 19 '18 at 17:35
  • The term was introduced by Maurice Frechet in 1904, at least according to Wikipedia. Perhaps someone who can read French can shed some light on the reasons for the terminology, if such were given. – Asaf Karagila Aug 19 '18 at 17:38
  • 1
    I dont know if it is the historic origin, but @JoséCarlosSantos do give an acceptable explanation. @ AsafKaragila – X Leo Aug 19 '18 at 17:40
  • That is good, XLiu, that you found an acceptable answer. Hold off on accepting it if you hope that others may also post an answer in an effort to explain it differently. (As a rule, once you accept an answer, the chances of getting any additional explanations plummets.) So no need to rush to accept the first answer you receive, unless/until others also post answers, and you still prefer the first one. – amWhy Aug 19 '18 at 17:43
  • Definitely. Hope there are many more interesting answers.@amWhy – X Leo Aug 19 '18 at 17:46

1 Answers1

3

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.

José Carlos Santos
  • 427,504