1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Hausdorff topology on five-element set that is not the discrete top.

  1. Jan 29, 2013 #1
    1. The problem statement, all variables and given/known data

    The textbook exercise asks for a Hausdorff topology on [itex] \{a,b,c,d,e\}[/itex] which is not the discrete topology (the power set). It is from "Introduction to Topology, Pure and Applied", by Adams and Franzosa.

    2. Relevant equations

    Let X be a set.

    Definition of topology (top for short): must include X and the empty set, must include all intersections of finitely many sets in the top, must include all unions of any sets in the top.

    Definition of Hausdorff: for any two elements x and y of X, there must be disjoint open sets in the top such that one contains x and the other contains y.

    3. The attempt at a solution

    My thought is that this cannot be done. We have a theorem that if X is Hausdorff, then every single point subset of X is closed. This implies that all the four-element subsets of X are (in the topology) and open. But then the intersection of any of the four-element sets generates the rest of the powerset.
  2. jcsd
  3. Jan 29, 2013 #2
    That is correct. Any finite Hausdorff space must be discrete!

    Your proof is correct as well!
  4. Jan 29, 2013 #3
    Thanks micro :)
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Hausdorff topology on five-element set that is not the discrete top.
  1. Discrete topology? (Replies: 1)