On all numbers great and small (Topological fields of Conway's numbers and their completions)

The proper Class No of all Conway's numbers l3 is considered as a region of investigation. It turns out to be a total ordered Field (i.e., a field whose domain is a proper Class) and this totally, or linear ordered Class, containing the real numbers R and the ordinal numbers { On}. For any subfield F of No, i.e., F is a set nor proper class, considered with topology induced by a linear ordering on F a completion F is constructed; in particular, for ζ=ωω^μ, 0≤μ<Ω, and for a specially defined subfield F= P_ζ⊂ No a complete subfield R_ζ⊂ No is defined as P̃_ζ. Fundamental (Cauchy) sequences (x_α)0≤α<ζ are considered in a subfield F⊂ P_ζ⊂ No, where ζ is the smallest ordinal number which does not belong to F, and they are the main instrument in the paper. A fragment of Mathematical Analysis in R_ζ is given and two of its non-trivial results are presented: every positive number x∈ R_ζ has a unique n-th root in R_ζ, for each positive integer n and every odd-degree polynomial with coefficients in R_ζ has a root in R_ζ. Hence so-called fundamental theorem of algebra: the ring R_ζ[i]def= C_ζ of all numbers of the form $x+iy$ (x,y∈ R_ζ), i²=-1, is an algebraically closed field.