Geodesic
Definability of 1-Types in Weakly $o$-Minimal Theories
Matematičeskie trudy, Tome 8 (2005) no. 2, pp. 3-38

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In the article, we prove a criterion for definability of 1-types over sets in weakly $o$-minimal theories in terms of left and right convergences of a formula to a type. Van den Dries proved that every type over the field of reals is definable. Marker and Steinhorn strengthened his result. They (and, later, Pillay) proved the following assertion. Let $M\prec N$ be a pair of models of some $o$-minimal theory. If, for each element of $N$, the type of this element over $M$ is definable then, for each tuple of elements of $N$, the type of this tuple over $M$ is definable. We construct a weakly $o$-minimal theory for which the Marker–Steinhorn theorem fails; i. e., some pair of models of the theory possesses the following property: For all elements of the larger model, the 1-type over the smaller model is definable but there exists a tuple of elements of the larger model whose 2-type over the smaller model is not definable. 
@article{MT_2005_8_2_a0,
     author = {B. S. Baizhanov},
     title = {Definability of {1-Types} in {Weakly} $o${-Minimal} {Theories}},
     journal = {Matemati\v{c}eskie trudy},
     pages = {3--38},
     publisher = {mathdoc},
     volume = {8},
     number = {2},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MT_2005_8_2_a0/}
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B. S. Baizhanov. Definability of 1-Types in Weakly $o$-Minimal Theories. Matematičeskie trudy, Tome 8 (2005) no. 2, pp. 3-38. http://geodesic.mathdoc.fr/item/MT_2005_8_2_a0/