- #1
Gear300
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Problem Statement:
Show that the set X of all ordinals less than the first uncountable ordinal is countably compact but not compact.
Let μ be the first uncountable ordinal.
The latter question is easy to show, but I stumbled upon a curiosity while attempting the former. In showing the former, I simply tried to show that every infinite subset of X should have a limit point (or in particular, an ω-accumulation point) in X. And so, in doing this, I needed to ensure that any infinite subset with μ as a limit point has another limit point in X. I reasoned that the first ω ordinals of this subset should only span a countable range of ordinals, since each of their co-initials are countable and a countable union of countable sets is countable. Any neighborhood of μ, however, is uncountable, so the limit point of the first ω ordinals of this subset cannot be μ. But when I considered the following set -
The sequence {ωn} = ω, ω2, ... , ωn, ...
- it was hard to discern a limit point other than ωω. Aside from what the exact nature of the first uncountable ordinal is chosen to be, there should still be a limit point somewhere before ωω. So simply put, what ordinals exist between ωω and the sequence I presented?
Show that the set X of all ordinals less than the first uncountable ordinal is countably compact but not compact.
Let μ be the first uncountable ordinal.
The latter question is easy to show, but I stumbled upon a curiosity while attempting the former. In showing the former, I simply tried to show that every infinite subset of X should have a limit point (or in particular, an ω-accumulation point) in X. And so, in doing this, I needed to ensure that any infinite subset with μ as a limit point has another limit point in X. I reasoned that the first ω ordinals of this subset should only span a countable range of ordinals, since each of their co-initials are countable and a countable union of countable sets is countable. Any neighborhood of μ, however, is uncountable, so the limit point of the first ω ordinals of this subset cannot be μ. But when I considered the following set -
The sequence {ωn} = ω, ω2, ... , ωn, ...
- it was hard to discern a limit point other than ωω. Aside from what the exact nature of the first uncountable ordinal is chosen to be, there should still be a limit point somewhere before ωω. So simply put, what ordinals exist between ωω and the sequence I presented?
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