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Math Help - Finding the closure of sets in a given topology

  1. #1
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    Finding the closure of sets in a given topology

    Hi guys,

    I have to find the closure of K = \left\{ \frac{1}{n} : n\in\mathbb{N} \right\} , and A=(0,1) in various topologies.

    1. In the standard topology, I think \overline{K} = K \cup \{0\}, and \overline{A} =[0,1]

    2. Now, for the finite complement topology, I'm unsure for both K and A.

    3. The upper limit topology, I'm also unsure for both K and A.

    4. The topology with basis \left\{(-\infty, a) : a\in\mathbb{R} \right\} I think for both K and A is the set of all positive real numbers.

    Thanks alot in advance,

    HTale.
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  2. #2
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    Feb 2009
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    Closure

    1. You are correct.

    2. Take any point x in R and fix an open set U containing it. Since the complement of U is finite it contains at least one member of each of K and A. Hence the closure of K is the whole of R, as is the closure of A.

    3. Not quite sure what you mean by upper limit topology, but I think it is the topology generated by open sets of the form (a,b]. On this assumption, the closures are, for K, K itself, and for A, (0,1].

    4. This gives the set of all non-negative real numbers as the closure of both K and A.
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