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Homework Help: Define a new topology on the reals

  1. Apr 9, 2010 #1
    1. The problem statement, all variables and given/known data
    Verify that taking [tex]\mathbb{R}[/tex], the empty set and finite sets to be closed gives a topology.

    2. Relevant equations

    3. The attempt at a solution

    Clearly the empty set is finite as it has 0 elemnts, and so is closed.

    If [tex] X_i [/tex], for i= {1,...,n}, are finite sets then clearly the union of finitely many finite sets is again finite and so is closed.

    Let [tex]|X_i| = m_i [/tex], where [tex]m_i[/tex] is the number of elements in [tex]X_i[/tex], then [tex]|\bigcap X_i |[/tex] is at most max{[tex]m_i[/tex]} or at least 0 if the intersection is empty. Either way it is again finite and so is defined as closed.

    I think I have shown the empty set, arb unions, arb intersections are closed in this topology, but I can't see how [tex] \mathbb{R}[/tex] could be included...
  2. jcsd
  3. Apr 9, 2010 #2
    Reread the description of the closed sets. It says:
    1) That all finite sets are closed.
    2) That the empty set is closed (this is really redundant, but there's no harm in including it).
    3) That [itex]\mathbb{R}[/itex] is closed.
  4. Apr 9, 2010 #3
    Of course, as defined in the statement of the question!!!

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