Sheng Li (gmachine1729) wrote,
Sheng Li
gmachine1729

Another characterization of compactness

Originally published at 狗和留美者不得入内. You can comment here or there.

The canonical definition of compactness of a topological space X is every open cover has finite sub-cover. We can via contraposition translate this to every family of open sets with no finite subfamily that covers X is not a cover. Not a cover via de Morgan’s laws can be characterized equivalently as has complements (which are all closed sets) which have finite intersection. The product is:

A topological space is compact iff for every family of closed sets with the finite intersection property, the intersection of that family is non-empty.

Tags: geometry/topology, point-set topology
Subscribe

  • Disqus comment search service released!

    Originally published at 狗和留美者不得入内. You can comment here or there. THIS HAS BEEN UPDATED. SEE…

  • Ron Maimon on Disqus

    Originally published at 狗和留美者不得入内. You can comment here or there. I used my Disqus comment search to get all of Ron Maimon’s comments on…

  • Found my Disqus search on Google

    Originally published at 狗和留美者不得入内. You can comment here or there. On the third or second or seventh page. I think I was a bit harsh on them lol.…

  • Post a new comment

    Error

    Anonymous comments are disabled in this journal

    default userpic

    Your reply will be screened

  • 0 comments