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Defining Topological Spaces help

  1. Feb 28, 2012 #1
    1. The problem statement, all variables and given/known data

    Let ℝ be set of real numbers. Which of the following collection of subsets of ℝ defines a topology in ℝ.

    a) The empty set and all sets which contain closed interval [0,1] as a subset.

    b)R and all subsets of closed interval [0,1].

    c)The empty set, ℝ and all sets such that A not subset of [0,1] and [0,1] not subset of A.

    Determine if obtained topology is connected and Hausdorff.

    3. The attempt at a solution

    Im not sure how to interpret subsets of the closed interval [0,1] and this doesnt seem like it would be an open set.
     
  2. jcsd
  3. Feb 28, 2012 #2

    HallsofIvy

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    "subset of [0, 1]" means exactly what it says- things like [1/2, 3/4], (1/3, 4/5), etc. This question does not ask it they are "open sets" in terms of the "usual topology", it asks whether the collection of all sets forms a topology.

    You should recall that to be a "topology" a collection of subsets of set X must
    1) contain X itself.
    2) contain the empty set
    3) contain the union of any sub-collection of these sets
    4) contain the intersection of any finite sub-colection of these sets
     
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