## Rocky Mountain Journal of Mathematics

### The structure of spaces of $\mathbb{R}$-places of rational function fields over real closed fields

Katarzyna Kuhlmann

#### Abstract

For arbitrary real closed fields $R$, we study the structure of the space $M(R(y))$ of $\mathbb{R}$-places of the rational function field in one variable over $R$ and determine its dimension to be 1. We determine small subbases for its topology and discuss a suitable metric in the metrizable case. In the case of non-archimedean $R$, we exhibit the rich variety of homeomorphisms of subspaces that can be found in such spaces.

#### Article information

Source
Rocky Mountain J. Math., Volume 46, Number 2 (2016), 533-557.

Dates
First available in Project Euclid: 26 July 2016

https://projecteuclid.org/euclid.rmjm/1469537476

Digital Object Identifier
doi:10.1216/RMJ-2016-46-2-533

Mathematical Reviews number (MathSciNet)
MR3529082

Zentralblatt MATH identifier
1380.12009

Keywords
Real places spaces of real places

#### Citation

Kuhlmann, Katarzyna. The structure of spaces of $\mathbb{R}$-places of rational function fields over real closed fields. Rocky Mountain J. Math. 46 (2016), no. 2, 533--557. doi:10.1216/RMJ-2016-46-2-533. https://projecteuclid.org/euclid.rmjm/1469537476

#### References

• 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.
• W.J. Charatonik and A. Dilks, On self-homeomorphic spaces, Topol. Appl. 55 (1994), 215–238.
• J. Dugundji, Topology, Allyn and Bacon, Boston, 1966.
• A.J. Engler and A. Prestel, Valued fields, Springer Monogr. Math., Springer-Verlag, Berlin, 2005.
• J. Harman, Chains of higher level orderings, Contemp. Math. 8 (1982), 141–174.
• 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.
• 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.
• 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.
• T.Y. Lam, Orderings, valuations and quadratic forms, CBMS Reg. Conf. 52, published for the Conf. Board Math. Sciences, Washington, 1983.
• J. Nagata, Modern dimension theory, Bibliotheca Math. 6, Interscience Publishers, John Wiley & Sons, Inc., New York, 1965.
• 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.