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Let X:=R^n with the Euclidean Topology. Is X first countable? Find a nested basis

  1. Sep 15, 2012 #1
    1. The problem statement, all variables and given/known data
    Let X:=ℝn with the Euclidean Topology. Is X first countable? Find a nested neighborhood basis for X at 5.


    2. Relevant equations
    If X is a topological space and p[itex]\in[/itex]X, a collection [itex]B[/itex]p of neighborhoods of p is called a neighborhood basis for X at p if every neighborhood of p contains some B[itex]\in[/itex][itex]B[/itex]p.

    We say X is first countable if there exists a countable neighborhood basis at each point.


    3. The attempt at a solution
    I say yes.
    Let p[itex]\in[/itex]X, the set of open balls Br(p) for r being rational forms a neighborhood basis at p. (That is, for all neighborhoods U of p, there is a Br(p)[itex]\subseteq[/itex]U)
    Since p was arbitrary and this [itex]B[/itex]p is countable (since rationals are countable), X is first countable.

    As well, we can just let the nest interval be defined as: B(1/2)i(5) for i being a natural number. Thus, B(1/2)i+1(5)<B(1/2)i(5).

    I am struggling a bit at this level of proof honestly, and I'm trying to stay afloat. Thank you!
     
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  3. Sep 15, 2012 #2

    jbunniii

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    Re: Let X:=R^n with the Euclidean Topology. Is X first countable? Find a nested bas

    Your proof looks fine. Just a few nitpicking details: first, [itex]r[/itex] needs to be rational AND positive. Second, although it's pretty obvious, you might say a few words about why you can find an [itex]r[/itex] such that [itex]B_r(p) \subseteq U[/itex].

    Your nested neighborhood basis at 5 is fine. Just one minor detail: instead of <, you want [itex]\subset[/itex].
     
  4. Sep 15, 2012 #3
    Re: Let X:=R^n with the Euclidean Topology. Is X first countable? Find a nested bas

    Thank you!

    Wouldn't it just be based on the density of the rationals?
    Also, thank you for catching my typo :)
     
  5. Sep 15, 2012 #4

    dextercioby

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    Re: Let X:=R^n with the Euclidean Topology. Is X first countable? Find a nested bas

    Yes, that is correct.
     
  6. Sep 16, 2012 #5
    Re: Let X:=R^n with the Euclidean Topology. Is X first countable? Find a nested bas

    Thanks again! This site is amazing!
     
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