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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


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    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.
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