Editing Order topology (section)

== Topology and ordinals ==<!-- This section is linked from [[Ordinal number]] -->

=== Ordinals as topological spaces ===
Any [[ordinal number]] can be viewed as a topological space by endowing it with the order topology (indeed, ordinals are [[well-order]]ed, so in particular [[totally ordered]]). Unless otherwise specified, this is the usual topology given to ordinals. Moreover, if we are willing to accept a [[proper class]] as a topological space, then we may similarly view the class of all ordinals as a topological space with the order topology.

The set of [[limit point]]s of an ordinal ''α'' is precisely the set of [[limit ordinal]]s less than ''α''.  [[Successor ordinal]]s (and zero) less than ''α'' are [[isolated point]]s in ''α''.  In particular, the finite ordinals and ω are [[discrete space|discrete]] topological spaces, and no ordinal beyond that is discrete.  The ordinal ''α'' is [[compact space|compact]] as a topological space if and only if ''α'' is either a [[successor ordinal]] or zero.

The [[closed set]]s of a limit ordinal ''α'' are just the closed sets in the sense that we have already defined, namely, those that contain a limit ordinal whenever they contain all sufficiently large ordinals below it.

Any ordinal is, of course, an open subset of any larger ordinal.  We can also define the topology on the ordinals in the following [[recursion|inductive]] way: 0 is the empty topological space, ''α''+1 is obtained by taking the [[Compactification (mathematics)|one-point compactification]] of ''α'', and for ''δ'' a limit ordinal, ''δ'' is equipped with the [[direct limit|inductive limit]] topology. Note that if ''α'' is a successor ordinal, then ''α'' is compact, in which case its one-point compactification ''α''+1 is the [[disjoint union]] of ''α'' and a point.

As topological spaces, all the ordinals are [[Hausdorff space|Hausdorff]] and even [[normal space|normal]].  They are also [[totally disconnected]] (connected components are points), [[scattered space|scattered]] (every non-empty subspace has an isolated point; in this case, just take the smallest element), [[zero-dimensional space|zero-dimensional]] (the topology has a [[clopen]] [[basis (topology)|basis]]: here, write an open interval (''β'',''γ'') as the union of the clopen intervals (''β'',''γ''<nowiki>'</nowiki>+1) = <nowiki>[</nowiki>''β''+1,''γ''<nowiki>']</nowiki> for ''γ''<nowiki>'</nowiki><''γ'').  However, they are not [[extremally disconnected]] in general (there are open sets, for example the even numbers from ω, whose [[closure (mathematics)|closure]] is not open).

The topological spaces ω<sub>1</sub> and its successor ω<sub>1</sub>+1 are frequently used as textbook examples of uncountable topological spaces. For example, in the topological space ω<sub>1</sub>+1, the element ω<sub>1</sub> is in the closure of the subset ω<sub>1</sub> even though no sequence of elements in ω<sub>1</sub> has the element ω<sub>1</sub> as its limit: an element in ω<sub>1</sub> is a countable set; for any sequence of such sets, the union of these sets is the union of countably many countable sets, so still countable; this union is an upper bound of the elements of the sequence, and therefore of the limit of the sequence, if it has one.

The space ω<sub>1</sub> is [[first-countable space|first-countable]] but not [[second-countable space|second-countable]], and ω<sub>1</sub>+1 has neither of these two properties, despite being [[compact space|compact]].  It is also worthy of note that any [[continuous function]] from ω<sub>1</sub> to '''R''' (the [[real line]]) is eventually constant: so the [[Stone–Čech compactification]] of ω<sub>1</sub> is ω<sub>1</sub>+1, just as its one-point compactification (in sharp contrast to ω, whose Stone–Čech compactification is much ''larger'' than ω).

=== Ordinal-indexed sequences ===
If ''α'' is a limit ordinal and ''X'' is a set, an ''α''-indexed sequence of elements of ''X'' merely means a function from ''α'' to ''X''.  This concept, a '''transfinite sequence''' or '''ordinal-indexed sequence''', is a generalization of the concept of a [[sequence]]. An ordinary sequence corresponds to the case ''α'' = ω.

If ''X'' is a topological space, we say that an ''α''-indexed sequence of elements of ''X'' ''converges'' to a limit ''x'' when it converges as a [[net (mathematics)|net]], in other words, when given any [[neighborhood (mathematics)|neighborhood]] ''U'' of ''x'' there is an ordinal ''β'' < ''α'' such that ''x''<sub>''ι''</sub> is in ''U'' for all ''ι'' ≥ ''β''.

Ordinal-indexed sequences are more powerful than ordinary (ω-indexed) sequences to determine limits in topology: for example, ω<sub>1</sub> is a limit point of ω<sub>1</sub>+1 (because it is a limit ordinal), and, indeed, it is the limit of the ω<sub>1</sub>-indexed sequence which maps any ordinal less than ω<sub>1</sub> to itself: however, it is not the limit of any ordinary (ω-indexed) sequence in ω<sub>1</sub>, since any such limit is less than or equal to the union of its elements, which is a countable union of countable sets, hence itself countable.

However, ordinal-indexed sequences are not powerful enough to replace nets (or [[filter (mathematics)|filter]]s) in general: for example, on the [[Tychonoff plank]] (the product space <math>(\omega_1+1)\times(\omega+1)</math>), the corner point <math>(\omega_1,\omega)</math> is a limit point (it is in the closure) of the open subset <math>\omega_1\times\omega</math>, but it is not the limit of an ordinal-indexed sequence.