Order Topology - Ordinal Space

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