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Topic: Subspaces of limit ordinals
Replies: 8   Last Post: Jul 1, 2014 11:02 AM

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William Elliot

Posts: 1,434
Registered: 1/8/12
Re: Subspaces of limit ordinals
Posted: Jun 30, 2014 11:40 PM
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On Mon, 30 Jun 2014, Herman Rubin wrote:
> On 2014-06-30, William Elliot <marsh@panix.com> wrote:

> > Let eta be a limit ordinal and A a subspace of eta.
>
> > Is the following correct?
> > A is homeomorphic to eta iff A is an unbounded, closed subset.

>
> Here is a counterexample. Let eta = omega^2, omega being the order
> type of the rationals. Then the sequence omega*n, n \in omega, is
> unbounded and closed, but is homeomorphic to omega, not eta.


As neither omega nor eta is an ordinal, how is that a counter example

Here's a counter example.
Let eta = omega_0 + omega_0 and A = { omega_0 + n | n in omega_0 }
Unbounded, closed A homeomorphic to omega_0.

> The theorem is true if eta is a cardinal, not the sum of a smaller
> number of cardinals.


Here's a counter example.
Let eta = aleph_(omega_0), A = { aleph_n | n in omega_0 }
Unbounded, closed A homeomorphic to aleph_0.

Is the theorem true only for regular cardinals?

On the other hand, is there a counter example to the proposition
that if A (subset eta) is homeomorphic to (limit ordinal) eta
then A is an unbounded, closed subset.

Yes, eta = omega_1, A = eta - {omega_0} which isn't closed.

Thus what is the theroem? Is this it?
If kappa is a regular cardinal and A an unbounded, closed
subset of kappa, then A is homeomorphic to kappa.



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