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Order-preserving injection

  1. Dec 18, 2007 #1
    1. The problem statement, all variables and given/known data
    Show that there does not exist an order-preserving injection from the ordinal [tex]\omega_1[/tex] to the reals (given the usual order).

    3. The attempt at a solution
    Suppose such an injection exists. Then something bad happens. Maybe the fact that w1 is well-ordered?
  2. jcsd
  3. Dec 18, 2007 #2


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    Give w_1 and R their order topologies. Then an order-preserving injection from w_1 into R is a topological embedding. But w_1 is not second countable, while R is. Contradiction, because a subspace of a second countable space is second countable.
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