Editing Order topology (section)

==Ordinal space==

For any [[ordinal number]] ''&lambda;'' one can consider the spaces of ordinal numbers
:<math>[0,\lambda) = \{\alpha \mid \alpha < \lambda\}</math>
:<math>[0,\lambda] = \{\alpha \mid \alpha \le \lambda\}</math>
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 ''&lambda;'' = [0, ''&lambda;'') and ''&lambda;'' + 1 = [0, ''&lambda;'']). Obviously, these spaces are mostly of interest when ''&lambda;'' is an infinite ordinal; for finite ordinals, the order topology is simply the [[discrete topology]].

When ''&lambda;'' = &omega; (the first infinite ordinal), the space [0,&omega;) is just '''N''' with the usual (still discrete) topology, while [0,&omega;] is the [[Alexandroff_extension|one-point compactification]] of '''N'''.

Of particular interest is the case when ''&lambda;'' = &omega;<sub>1</sub>, the set of all countable ordinals, and the [[first uncountable ordinal]]. The element &omega;<sub>1</sub> is a [[limit point]] of the subset [0,&omega;<sub>1</sub>) even though no [[sequence]] of elements in [0,&omega;<sub>1</sub>) has the element &omega;<sub>1</sub> as its limit. In particular, [0,&omega;<sub>1</sub>] is not [[First-countable space|first-countable]]. The subspace [0,&omega;<sub>1</sub>) is first-countable however, since the only point in [0,&omega;<sub>1</sub>] without a countable [[local base]] is &omega;<sub>1</sub>. Some further properties include
*neither [0,&omega;<sub>1</sub>) or [0,&omega;<sub>1</sub>] is [[separable space|separable]] or [[second-countable]]
*[0,&omega;<sub>1</sub>] is [[compact space|compact]], while [0,&omega;<sub>1</sub>) is [[Sequentially compact space|sequentially compact]] and [[Countably compact space|countably compact]], but not compact or [[paracompact]]