Ordinal Space
For any ordinal number λ one can consider the spaces of ordinal numbers
together with the natural order topology. These spaces are called ordinal spaces. (Note that in the usual set-theoretic construction of ordinal numbers we have λ = ). Obviously, these spaces are mostly of interest when λ is an infinite ordinal; otherwise (for finite ordinals), the order topology is simply the discrete topology.
When λ = ω (the first infinite ordinal), the space is the one-point compactification of N.
Of particular interest is the case when λ = ω1, the set of all countable ordinals, and the first uncountable ordinal. The element ω1 is a limit point of the subset is not first-countable. The subspace [0,ω1) is first-countable however, since the only point without a countable local base is ω1. Some further properties include
- neither is separable or second-countable
- is compact while [0,ω1) is sequentially compact and countably compact, but not compact or paracompact
Read more about this topic: Order Topology
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