## The structure of spaces of $$\mathbb{R}$$-places of rational function fields over real closed fields.(English)Zbl 1380.12009

For a formally real field $$K$$, denote by $$\mathcal{X}(K)$$ the compact, Hausdorff, totally disconnected (i.e. Boolean) space of orderings of $$K$$ endowed with the Harrison topology and let $$M(K)$$ be the set of real places $$\xi: K\rightarrow \bar \mathbb{R}=\mathbb{R}\cup \{\infty\}$$. There is a natural surjection $$\lambda: \mathcal{X}(K) \rightarrow M(K)$$ and one endows $$M(K)$$ with the quotient topology (here also called Harrison topology) subordinated to this map. See [E. Becker, Lect. Notes Math. 959, 1–40 (1982; Zbl 0508.14013)].
The author’s main concern is the case of the rational function field $$K=R(y),$$ with $$R$$ real closed, which we assume throughout. If $$R$$ is non-Archimedean, it is found, in her words, that spaces $$M(R(y))$$ have an amazingly rich topological structure.
Section 1 introduces the ultrametric on $$R$$ defined by $$u(a,b)=v(a-b)$$, where $$v$$ is the natural valuation coming from the unique ordering of $$R$$. For any cut $$(S,T)$$ in the value group $$vR$$, the ultrametric balls of center $$a$$ and radius $$T$$, $$B_T(a)=\{b\in R: u(a,b)\in T\cup \infty \}$$ are defined; they play a pivotal rôle in the study. One important tool is the known Proposition 1.1: The set of cuts $$\mathcal{C}(R)$$ endowed with the order topology and the space $$\mathcal{X}(R(y))$$ with the Harrison topology are homoeomorphic via a map $$\chi$$.
Section 2 finds a small subbasis for the topological space $$M(R(y))$$. For $$f\in R(y),$$ let $$U(f)=\{\xi\in M(R(y)): \xi(f)\in \mathbb{R}^+\}$$. Theorem 2.5: Suppose $$F$$ is a field dense in $$R$$. Consider the family $$\mathcal{F}=\{a+by, \frac{y-a}{y-b}: a,b \in F\}.$$ Then the family $$U(f),$$ $$f\in \mathcal{F}$$ forms a subbasis for the Harrison topology on $$M(R(y))$$.
Two proper subfamilies of $$\mathcal{F}$$ are identified that separate points in $$M(R(y))$$ in a simple and in a stronger topological sense, respectively. The space $$M(R(y))$$ is metrizable iff there is countable real closed field $$F$$ which is dense in $$R,$$ see F.-V. Kuhlmann et al. [Commun. Algebra 39, No. 9, 3166–3177 (2011; Zbl 1263.12001)]. This metric is now described. Take any bijection $$\sigma: \mathcal{F} \rightarrow \mathbb{N}$$. Let $$d_0$$ be any fixed metric on $$\bar \mathbb{R}$$. Theorem 3.3: The following mapping $$d: M(R(y))\times M(R(y)) \rightarrow [0,\infty[$$ is a metric on $$M(R(y))$$: $$d(\xi,\eta)=\sup_{f\in \mathcal{F}} \{2^{-\sigma(f)} d_0(\xi(f),\eta(f))\}$$.
A theorem in J. Nikiel et al. [Pac. J. Math. 149, 145–155 (1991; Zbl 0687.54023)] says that for the continuous image of a compact ordered space, the covering dimension, the small inductive dimension, and the large inductive dimension are all equal. Since by section 1, $$M(R(y))$$ can be seen as image of the compact space $$\mathcal{C}(R)$$, some additional argument using connectedness of $$M(R(y)),$$ see [J. Harman, Contemp. Math. 8, 141–174 (1982; Zbl 0509.12021)], yields the main result of section 4. Theorem: $$1=\dim M(R(y))=\text{ind} M(R(y))= \text{Ind} M(R(y)).$$
Section 5 studies homeomorphisms on $$M(R(y))$$. Using that the automorphisms of $$R(y)$$ that leave $$R$$ elementwise fixed are the Möbius transformations $$y\mapsto (ay+b)/(cy+d)$$ with $$ad-bc\neq 0$$ and that these can be written as compositions of translations, multiplication, and inversion, it is first shown in Proposition 5.1 that these actions induce homeomorphisms on $$\mathcal{C}(R)$$ which are compatible with equivalence of cuts (defined earlier by means of the ultrametric balls). By the composition $$\lambda \circ\chi$$ from $$\mathcal{C}(R)$$ to $$M(R(y))$$ the author gets results for $$M(R(y))$$, in particular cofinal and coinitial chains of subspaces of $$M(R(y))$$ which are all mutually homeomorphic.
The field $$R=\mathbb{R}((t^{\mathbb{Q}}))$$ is known to be real closed. As the final result of section 5, inspired partially by W. Charatonik and A. Dilks [Topology Appl. 55, No. 3, 215–238 (1994; Zbl 0788.54040)], it is shown in Corollary 5.4 that for this $$R,$$ the space $$M(R(y))$$ has the remarkable property that every open subset contains a homeomorphic copy of $$M(R(y))$$; so $$M(R(y))$$ is ‘self-homoeomorphic’.
The final section 6 describes the big picture of what the author calls – apparently with good reasons – the fractal structure of $$M(R(y))$$ in case of non-archimedean $$R$$. Future papers detailing this are promised.

### MSC:

 12J15 Ordered fields 12J25 Non-Archimedean valued fields 12J20 General valuation theory for fields 13J30 Real algebra 54E45 Compact (locally compact) metric spaces 54F45 Dimension theory in general topology
Full Text:

### References:

 [1] T. Banakh, Ya. Kholyavka, K. Kuhlmann, M. Machura and O. Potyatynyk, The dimension of the space of $$\R$$-places of certain rational function fields , Cent. Europ. J. Math. 12 (2014), 1239-1248. · Zbl 1311.12004 · doi:10.2478/s11533-014-0409-y [2] W.J. Charatonik and A. Dilks, On self-homeomorphic spaces , Topol. Appl. 55 (1994), 215-238. · Zbl 0788.54040 · doi:10.1016/0166-8641(94)90038-8 [3] J. Dugundji, Topology , Allyn and Bacon, Boston, 1966. [4] A.J. Engler and A. Prestel, Valued fields , Springer Monogr. Math., Springer-Verlag, Berlin, 2005. · Zbl 1128.12009 · doi:10.1007/3-540-30035-X [5] J. Harman, Chains of higher level orderings , Contemp. Math. 8 (1982), 141-174. · Zbl 0509.12021 · doi:10.1090/conm/008/653181 [6] F.-V. Kuhlmann and K. Kuhlmann, Embedding theorems for spaces of $$\R$$-places of rational function fields and their products , Fund. Math. 218 (2012), 121-149. · Zbl 1262.12002 · doi:10.4064/fm218-2-2 [7] F.-V. Kuhlmann, S. Kuhlmann, M. Marshall and M. Zekavat, Embedding ordered fields in formal power series fields , J. Pure Appl. Algebra 169 (2002), 71-90. · Zbl 0998.12010 · doi:10.1016/S0022-4049(01)00064-0 [8] F.-V. Kuhlmann, M. Machura and K. Osiak, Spaces of $$\mathbb R$$-places of function fields over real closed fields , Comm. Algebra 39 (2011), 3166-3177. · Zbl 1263.12001 · doi:10.1080/00927872.2010.496752 [9] T.Y. Lam, Orderings, valuations and quadratic forms , CBMS Reg. Conf. 52 , published for the Conf. Board Math. Sciences, Washington, 1983. [10] J. Nagata, Modern dimension theory , Bibliotheca Math. 6 , Interscience Publishers, John Wiley & Sons, Inc., New York, 1965. · Zbl 0129.38304 [11] J. Nikiel, H.M. Tuncali and E.D. Tymchatyn, On the rim-structure of continuous images of ordered compacta , Pac. J. Math. 149 (1991), 145-155. · Zbl 0687.54023 · doi:10.2140/pjm.1991.149.145
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.