## Notre Dame Journal of Formal Logic

### Tame Topology over dp-Minimal Structures

#### Abstract

In this article, we develop tame topology over dp-minimal structures equipped with definable uniformities satisfying certain assumptions. Our assumptions are enough to ensure that definable sets are tame: there is a good notion of dimension on definable sets, definable functions are almost everywhere continuous, and definable sets are finite unions of graphs of definable continuous “multivalued functions.” This generalizes known statements about weakly o-minimal, C-minimal, and P-minimal theories.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 60, Number 1 (2019), 61-76.

Dates
Accepted: 15 November 2016
First available in Project Euclid: 16 January 2019

https://projecteuclid.org/euclid.ndjfl/1547607904

Digital Object Identifier
doi:10.1215/00294527-2018-0019

Mathematical Reviews number (MathSciNet)
MR3911106

Subjects
Primary: 03C68: Other classical first-order model theory

#### Citation

Simon, Pierre; Walsberg, Erik. Tame Topology over dp-Minimal Structures. Notre Dame J. Formal Logic 60 (2019), no. 1, 61--76. doi:10.1215/00294527-2018-0019. https://projecteuclid.org/euclid.ndjfl/1547607904

#### References

• [1] Cubides-Kovacsics, P., L. Darnière, and E. Leenknegt, “Topological cell decomposition and dimension theory in $P$-minimal fields,” Journal of Symbolic Logic, vol. 82 (2017), pp. 347–58.
• [2] Dolich, A., J. Goodrick, and D. Lippel, “Dp-minimality: Basic facts and examples,” Notre Dame Journal of Formal Logic, vol. 52 (2011), pp. 267–88.
• [3] Eleftheriou, P., A. Hasson, and G. Keren, “On definable Skolem functions in weakly o-minimal non-valuational structures,” Journal of Symbolic Logic, vol. 82 (2017), no. 4, 1482–1495.
• [4] Goodrick, J., “A monotonicity theorem for dp-minimal densely ordered groups,” Journal of Symbolic Logic, vol. 75 (2010), pp. 221–38.
• [5] Jahnke, F., P. Simon, and E. Walsberg, “Dp-minimal valued fields,” Journal of Symbolic Logic, vol. 82 (2017), pp. 151–165.
• [6] Johnson, W., “The canonical topology on dp-minimal fields,” to appear in Journal of Mathematical Logic (2018).
• [7] Macpherson, D., D. Marker, and C. Steinhorn, “Weakly o-minimal structures and real closed fields,” Transactions of the American Mathematical Society, vol. 352 (2000), pp. 5435–83.
• [8] Simon, P., “On dp-minimal ordered structures,” Journal of Symbolic Logic, vol. 76 (2011), pp. 448–60.
• [9] Simon, P., “Dp-minimality: Invariant types and dp-rank,” Journal of Symbolic Logic, vol. 79 (2014), pp. 1025–45.
• [10] Simon, P., A Guide to NIP Theories, vol. 44 of Lecture Notes in Logic, Cambridge University Press, Cambridge, 2015.