Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Questions on locally compact space

  1. Oct 31, 2012 #1
    Q1. if A is a subset of X, choose the topology on X as {∅,U|for every U in X that A is a subset of U}. Then is this topology a locally compact space?
    Q2. X=[-1,1], the topology on X is {∅, X, [-1, b), (a,b), (a,1] | for all a<0<b}. How to prove every open set (a,b) in X is NOT locally compact?

    for Q1, there is so few restrictions on X, I don't know whether it's a locally compact space or not, however, I also cannot find a counterexample.
    for Q2, when defined open set of a specific topology, does is mean that we have also defined closed set? If so, then for every point a in X, a closed set including a is a compact subset, then it is locally compact, a contradiction.
     
  2. jcsd
  3. Nov 1, 2012 #2

    Bacle2

    User Avatar
    Science Advisor

    Usually, you find a condition satisfied by your space that is not satisfied by locally-compact spaces.

    Example : Rationals as a subspace of Reals are not locally compact; in metric spaces,

    compactness is equivalent to every sequence having a convergent subsequence.

    Then, e.g. the sequences given by:

    1) a1=1, a1=1.4 , a3=1.41 ,.... (first n terms of the decimal expansion of √2 )

    2) a1=1 , a2=2,..., an =n ,....

    are sequences without convergent subsequences. In non-metric spaces it is a little harder.

    Still, re #2 : I have never seen a description of local compactness in terms of open sets,

    but in terms of points. What definition are you using?
     
  4. Nov 1, 2012 #3
    I mean, in #2, that if every point in the open set (a,b) has a compact subset, then the open set is locally compact.
     
  5. Nov 1, 2012 #4
    in #1. the topology on X has already defined, which mean if choosing rationals as the subset A, then U are all the subsets including A, it is like a discrete topology, so imposing a metric topology on it seems not right.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Questions on locally compact space
  1. Locally compact (Replies: 3)

  2. Compact Spaces (Replies: 4)

Loading...