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Topology generation

  1. May 8, 2010 #1
    1. The problem statement, all variables and given/known data

    Let [tex](X,\tau)[/tex] be [tex]X = \mathbb{R}[/tex] equipped with the topology
    generated by [tex]\EuScript{E} := \{[a,\infty) | a \in \mathbb{R} \}[/tex].

    Show that [tex]\tau = \{ \varnothing, \mathbb{R} \} \cup \{
    [a,\infty), (a, \infty) | a \in \mathbb{R} \}[/tex]

    2. Relevant equations

    A topology generated by [tex]\EuScript{E}[/tex] is [tex]\tau(\EuScript{E}) = \bigcap \{ \tau \subset \mathcal{P}(X) | \tau \text{ is a topology } \wedge \tau \supset \EuScript{E} \}[/tex]

    3. The attempt at a solution

    I can see that [tex]\tau = \{ \varnothing, \mathbb{R} \} \cup \{
    [a,\infty), (a, \infty) | a \in \mathbb{R} \}[/tex] is a topology for [tex]X[/tex]. But I don't know why the generated topology contains [tex](a,\infty)[/tex] as well. How is this obtained? How should I prove that [tex]\tau = \{ \varnothing, \mathbb{R} \} \cup \{
    [a,\infty), (a, \infty) | a \in \mathbb{R} \}[/tex] is the intersection of all topologies containing [tex]\EuScript{E}[/tex]?
  2. jcsd
  3. May 8, 2010 #2


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    Science Advisor
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    Gold Member

    Consider the sets

    [tex]\left[a + \frac{1}{n}, \infty\right)[/tex]

    for [itex]n = 1,2,\ldots[/itex]

    What is the union of these sets?
  4. May 8, 2010 #3
    I see. The union of these sets is [tex](a,\infty)[/tex]. Hence [tex](a,\infty)[/tex] must be in the topology in order to satisfy the closed under arbitrary union condition.

    Thanks very much for your help.
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