# Order-preserving injection

1. Dec 18, 2007

### Dragonfall

1. The problem statement, all variables and given/known data
Show that there does not exist an order-preserving injection from the ordinal $$\omega_1$$ 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. Dec 18, 2007

### morphism

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.