We prove that the space $M(K(x,y))$ of $\mathbb R$-places of the field $K(x,y)$ of rational functions of two variables with coefficients in a totally Archimedean field $K$ has covering and integral dimensions $\dim M(K(x,y))=\dim_\IZ M(K(x,y))=2$ and the cohomological dimension $\dim_G M(K(x,y))=1$ for any Abelian 2-divisible coefficient group $G$.
In this paper we study the topological structure and evaluate the topological dimensions of the spaces of R-places of a field K and of its transcendental extensions K(x 1 , . . . , x n ) consisting of rational functions of n variables with coefficients in the field K.
The shortest possible way to introduce R-places on a field K is to define them as functions χ : K → R = R ∪ {∞} to the extended real line, preserving the arithmetic operations in the sense that χ(0) = 0, χ(1) = 1, χ(x + y) ∈ χ(x) ⊕ χ(y) and χ(x • y) ∈ χ(x) ⊙ χ(y) for all x, y ∈ K, where ⊕ and ⊙ are multivalued extensions of the addition and multiplication operations from R to R. By definition, for r, s ∈ R, r ⊕ s = {r + s} if r + s ∈ R is defined and r ⊕ s = R if r + s is not defined, which happens if and only if r = s = ∞, in which case ∞ ⊕ ∞ = R. By analogy, we define r ⊙ s: it equals the singleton {r • s} if r • s is defined and R in the other case, which happens if and only if {r, s} = {0, ∞}.
Historically, R-places appeared from studying ordered fields. By an ordered field we understand a pair (K, P ) consisting of a field K and a subset P ⊂ K called the positive cone of (K, P ) such that P is an additively closed subgroup of index 2 of the multiplicative group of K. There is a bijective correspondence between positive cones of K and linear orders compatible with addition and multiplication by positive elements. The set {a ∈ K : a > 0} is a positive cone, and the positive cone P generates a total order < on K defined by x < y ⇔ y -x ∈ P . Each ordered field (K, P ) has characteristic zero and hence contains the field Q of rational numbers as a subfield. This fact allows us to define the Archimedean part
of the ordered field (K, P ) and also to define the canonical R-place χ P : K → R on K assigning χ P (x) = ∞ to each x ∈ K \ A P (K) and
to each x ∈ A P (K). Here the supremum and infimum is taken in the ordered field R of real numbers.
According to Theorems 1 and 6 of [12], a field K admits a R-place if and only if it is orderable in the sense that it admits a total order. By [3], each R-place χ : K → R on a field K is generated by a suitable total order P on K.
For an orderable field K denote by X (K) the space of total orders on K and by M (K) the space of R-places on K. The mentioned results [12] and [3] imply that the map λ : X (K) → M (K), λ : P → χ P , assigning to each total order P on K the corresponding R-place χ P is surjective.
The spaces X (K) and M (K) carry natural compact Hausdorff topologies. Namely, X (K) carries the Harrison topology generated by the subbase consisting of the sets a + = {P ∈ X (K) : a ∈ P } where a ∈ K \ {0}. According to [7, 6.1], the space X (K) endowed with the Harrison topology is compact Hausdorff and zero-dimensional. By [4], each compact Hausdorff zero-dimensional space is homeomorphic to the space of orderings X (K) of some field K.
To introduce a natural topology on the space M (K) of R-places of a field K, first endow the extended real line R = R ∪ {∞} with the topology of one-point compactification of the real line R. It follows from the definition of R-places that the space M (K) is a closed subspace of the compact Hausdorff space RK of all functions from K to R, endowed with the topology of Tychonoff product of the circles R. So, M (K) is a compact Hausdorff space, being a closed subspace of the compact Hausdorff space RK .
It turns out that the topology induced on M (K) by the product topology coincides with the quotient topology induced by the mapping λ : X (K) → M (K). This can be seen as follows. By [10], the sets
compose a sub-basis of the quotient topology on M (K). Since those sets are open in the product topology of M (K), the quotient topology is weaker than the product topology. Since the quotient topology is Hausdorff (see [11,Cor.9.9]) and the product topology is compact (so the weakest among Hausdorff topologies), both topologies on M (K) coincide.
The space M (K) ⊂ RK is metrizable if the field K is countable. The converse statement is not true as the uncountable field R has trivial space of R-places M (R) = {id}. The space of R-places M (R(x)) of the field R(x) is homeomorphic to the projective line R while M (R(x, y)) is not metrizable, see [14].
In this paper we shall address the following general problem posed in [2].
Problem 1.1. Investigate the interplay between algebraic properties of a field K and topological properties of its space of R-places M (K).
We shall be mainly interested in the fields K(x 1 , . . . , x n ) of rational functions of n variables with coefficients in a subfield K ⊂ R. It is known that a field K is isomorphic to a subfield of R if and only if K admits an Archimedean order, i.e., a total ordering P whose Archimedean part A P (K) coincides with K. This happens if and only if the corresponding R-place χ P : K → R is injective if and only if χ P (K) ⊂ R. By M A (K) we denote the space of injective R-places on K. Observe that M A (K) coinc
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