Topological Properties
Any ordinal number can be turned into a topological space by using the order topology. When viewed as a topological space, ω1 is often written as [0,ω1) to emphasize that it is the space consisting of all ordinals smaller than ω1.
Every increasing ω-sequence of elements of [0,ω1) converges to a limit in [0,ω1). The reason is that the union (=supremum) of every countable set of countable ordinals is another countable ordinal.
The topological space [0,ω1) is sequentially compact but not compact. It is however countably compact and thus not Lindelöf. In terms of axioms of countability, [0,ω1) is first countable but not separable nor second countable. As a consequence, it is not metrizable.
The space = ω1 + 1 is compact and not first countable. ω1 is used to define the long line and the Tychonoff plank, two important counterexamples in topology.
Read more about this topic: First Uncountable Ordinal
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