Editing Surreal number (section)

==Gaps and continuity==

In contrast to the real numbers, a (proper) subset of the surreal numbers does not have a least upper (or lower) bound unless it has a maximal (minimal) element. Conway defines<ref name=Con01/> a gap as {{math|{{mset| ''L'' {{!}} ''R'' }}}} such that every element of {{math|''L''}} is less than every element of {{math|''R''}}, and <math display=inline> L \cup R = \mathbb{No}</math>; this is not a number because at least one of the sides is a proper class. Though similar, gaps are not quite the same as [[Dedekind cut]]s,{{efn|The definition of a gap omits the conditions of a Dedekind cut that {{math|''L''}} and {{math|''R''}} be non-empty and that {{math|''L''}} not have a largest element, and also the identification of a cut with the smallest element in {{math|''R''}} if one exists.}} but we can still talk about a completion <math display=inline>\mathbb{No}_\mathfrak{D}</math> of the surreal numbers with the natural ordering which is a (proper class-sized) [[linear continuum]].<ref name=RSS15>{{cite arXiv|first1=Simon|last1=Rubinstein-Salzedo|first2=Ashvin|last2=Swaminathan|eprint=1307.7392v3|class=math.CA|date=2015-05-19|title=Analysis on Surreal Numbers}}</ref>

For instance there is no least positive infinite surreal, but the gap

<math display=block>\{ x : \exists n \in \mathbb N : x < n\mid x : \forall n\in \mathbb N : x > n \}</math>

is greater than all real numbers and less than all positive infinite surreals, and is thus the least upper bound of the reals in <math display=inline>\mathbb{No}_\mathfrak{D}</math>. Similarly the gap <math display=inline>\mathbb{On} = \{ \mathbb{No} \mid{} \}</math> is larger than all surreal numbers. (This is an [[Mathematical joke|esoteric pun]]: In the general construction of ordinals, {{mvar|α}} "is" the set of ordinals smaller than {{mvar|α}}, and we can use this equivalence to write {{nowrap|{{math|''α'' {{=}} {{mset| ''α'' {{!}} }}}}}} in the surreals; <math display=inline>\mathbb{On}</math> denotes the class of ordinal numbers, and because <math display=inline>\mathbb{On}</math> is [[Cofinal (mathematics)|cofinal]] in <math display=inline>\mathbb{No}</math> we have <math display=inline> \{ \mathbb{No} \mid {} \} = \{ \mathbb{On} \mid {} \} = \mathbb{On}</math> by extension.)

With a bit of set-theoretic care,{{efn|Importantly, there is no claim that the collection of Cauchy sequences constitutes a class in NBG set theory.}} <math display=inline>\mathbb{No}</math> can be equipped with a topology where the [[open set]]s are unions of open intervals (indexed by proper sets) and continuous functions can be defined.<ref name=RSS15/> An equivalent of [[Cauchy sequence]]s can be defined as well, although they have to be indexed by the class of ordinals; these will always converge, but the limit may be either a number or a gap that can be expressed as
<math display=block>\sum_{\alpha\in\mathbb{No}} r_\alpha \omega^{a_\alpha}</math>
with {{math|''a''{{sub|''α''}}}} decreasing and having no lower bound in <math display=inline>\mathbb{No}</math>. (All such gaps can be understood as Cauchy sequences themselves, but there are other types of gap that are not limits, such as {{math|∞}} and <math display=inline>\mathbb{On}</math>).<ref name=RSS15/>