The Set of All Countable Ordinals, $\omega_1$
I'm trying to understand several things regarding $\omega_1$ and trying to get a better intuition. I have four questions regarding this space (in bold), but I also tried to summarize here what I know about the space, hoping that it helps anyone who comes across this post in the future. Thank you for any insights.
1) Definition
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$\omega_1$ denotes the set of all countable ordinals and $\{\omega_1\}$ is the first uncountable ordinal.
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Elements are all countable ordinals – I imagine the $\omega_1$ space as $0, 1, 2, 3, …. \omega, \omega + 1, \omega + 2, … \omega + \omega, … \omega * \omega, …\omega_1 $
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The element $\{ \omega_1 \}$ itself is not part of the space. It is the limit point.
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Topology is precisely the order topology, given by linear order $\leq$.
2) Properties
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$\omega_1$ is Hausdorff, locally compact
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Its cardinality is $\aleph_1$
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It is first-countable
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0 and every successor ordinal $\alpha+1$ are isolated points
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It is not compact (since the collection $\{ [0,\alpha):\alpha<\omega_1 \}$ is an open cover with no finite subcover.
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Not connected. (For instance, 0 (the least element) is an isolated point, which means that the set {0} is open. It is also closed (as is any finite set in ω1 or any other Hausdorff space) and so its complement is open. Thus we have written $\omega_1$ as a union of two nonempty disjoint open sets, so it is not connected.)
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Not separable
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Is it a discrete space?
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How do the open and closed subspaces look?
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Is it (strongly) zero-dimensional? How does the basis look?
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Are clopen subspaces homeomorphic to the whole space?
Best Answer
Minor correction: $\omega_1$ is both the first uncountable ordinal and the set of countable ordinals. $\{\omega_1\}$ is just a singleton set that contains it as its single element, so far from uncountable nor an ordinal.
I agree fully with all the properties.
Limit ordinals (so the non-succesor ones) like $\omega, \omega+\omega, \omega\cdot \omega, \omega^\omega$ (all in ordinal arithmetic) which are all elements of $\omega_1$ are not isolated points: e.g. the first limit $\omega$ has as its basic neighbourhoods $(n, \omega+1)=(n,\omega]$ (open interval, $n < \omega$ so $n$ a natural. Just open intervals containing $\omega$ (we can assume that the right hand boundary point is $\omega+1$ WLOG and the left hand boundary point of the interval must lie below $\omega$). All these are infinite. These intervals are also clopen, as is easy to see.
So $\omega_1$ is not discrete, there are $\aleph_1$ many limits in it, all of which are not isolated.
So basic elements are $\{\alpha\}$ for $\alpha =0$ or $\alpha$ a successor and are clopen. A basic element for a limit is of the form $(\beta, \alpha]$ where $\beta < \alpha$. All of these are clopen (complement is $[0,\beta+1) \cup (\alpha, \rightarrow)$ which is open too).
I wouldn't know an easy form of a general clopen set.
$\omega_1$ is (like all ordered spaces) hereditarily normal, aka $T_5$) and countably paracompact.
$\omega+1 = \omega_1 \cup \{\omega_1\}$ is compact and zero-dimensional and $\beta \omega_1 = \omega_1+1$ and so by a standard theorem $\omega_1$ is strongly zero-dimensional.
There are many clopen subsets that are merely countable and so not homeomorphic to $\omega_1$, e.g. $\omega+1$ and all isolated points (and all the basic sets I discussed). I think that $[\omega+1, \rightarrow) = \omega_1\setminus (\omega+1)$ is a clopen subset that is homeomorphic to the whole space.