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An NIP structure which does not interpret an infinite group but whose Shelah expansion interprets an infinite field [article]

Erik Walsberg
2019 arXiv   pre-print
The proof is complicated by the fact that a definable open set need not be a union of finitely many open cells.  ...  U is a finite union of regular open R-definable sets.  ... 
arXiv:1910.13504v1 fatcat:xsejcsy2m5eu5h45koqmyhb5dq

Structures having o-minimal open core

Alfred Dolich, Chris Miller, Charles Steinhorn
2009 Transactions of the American Mathematical Society  
By the Cell Decomposition Theorem, every set definable in an o-minimal structure is a Boolean combination of open definable sets.  ...  If T is an extension of the theory of densely ordered groups such that every open unary set definable in any model of T is a finite union of open intervals, then every model of T has o-minimal open core  ...  definable unary set is a finite union of open intervals.  ... 
doi:10.1090/s0002-9947-09-04908-3 fatcat:gqydu6uuezgandlqtgmd7r4cuu

Almost o-minimal structures and 𝔛-structures [article]

Masato Fujita
2021 arXiv   pre-print
We introduce the notion of multi-cells and demonstrate that any definable set is a finite union of multi-cells in the course of the proof of the above theorem.  ...  The former is a first-order expansion of a dense linear order without endpoints such that the intersection of a definable set with a bounded open interval is a finite union of points and open intervals  ...  In particular, the definable set ]c, d[∩D N is not a finite union of points and open intervals. Here, the notation D N be the subset of N defined by the same formula as D.  ... 
arXiv:2104.01312v1 fatcat:puwzoagquffhpphd2dsf5c7dbm

Locally o-minimal structures and structures with locally o-minimal open core

Antongiulio Fornasiero
2013 Annals of Pure and Applied Logic  
We study first-order expansions of ordered fields that are definably complete, and moreover either are locally o-minimal, or have a locally o-minimal open core.  ...  We give a characterisation of structures with locally o-minimal open core, and we show that dense elementary pairs of locally o-minimal structures have locally o-minimal open core.  ...  Thus, by Lemma 2.11, it is clear that X is the union of a pseudofinite set and an open cell, and therefore X is the union of 2 multi-cells.  ... 
doi:10.1016/j.apal.2012.10.002 fatcat:fmwztdzvvva35prhg3qunus77i

A theorem about topological $n$-cells

M. K. Fort
1954 Proceedings of the American Mathematical Society  
as the union of a finite number of members of 2; (3) SiEPi for each *'.  ...  It is easy to see that the closure of Sm -Pm~i can be expressed as the union of a finite set <ri, ■ ■ ■ , ap of elements of S.  ... 
doi:10.1090/s0002-9939-1954-0062434-9 fatcat:opns6p56xba3jeb5b7qzlgmgii

Extension of Lipschitz maps definable in Hensel minimal structures [article]

Krzysztof Jan Nowak
2022 arXiv   pre-print
It may be regarded as a definable, non-Archimedean, non-locally compact version of Kirszbraun's theorem.  ...  In this paper, we establish a theorem on extension of Lipschitz maps definable in Hensel minimal, non-trivially valued fields K of equicharacteristic zero.  ...  The union of B is a finite union of 0-definable reparametrized open cells C i .  ... 
arXiv:2204.05900v2 fatcat:mwp2gowz3vh2jily3q5m6agb4m

Topological invariance of the combinatorial Euler characteristic of tame spaces

Tibor Beke
2011 Homology, Homotopy and Applications  
We prove the topological invariance of the combinatorial Euler characteristic with the help of a canonical, topologically defined stratification of tame spaces by locally compact, tame strata.  ...  Any definable set X possesses finite decompositions into cylindrical cells and one can let eu S (X) = ∑ α∈cell(X) (−1) dim(α) , for any such decomposition.  ...  Let W be definable, locally compact and stratified into definable sets. Let X be a union of strata. Define top(X) = union of {α ⊆ X | for all β ⊆ W and γ ⊆ X, if α β γ, then β ⊆ X}.  ... 
doi:10.4310/hha.2011.v13.n2.a11 fatcat:ifdbwoyv75hkbde2y6xbidouqm

O-minimalism [article]

Hans Schoutens
2011 arXiv   pre-print
We propose a theory, DCTC (Definable Completeness/Type Completeness), that describes many properties of o-minimalistic structures (dimension theory, monotonicity, Hardy structures, quasi-cell decomposition  ...  Failure of cell decomposition leads to the related notion of a tame structure, and we give a criterium for an o-minimalistic structure to be tame.  ...  By Theorem 3.14, we can write Y as a disjoint union of open intervals and a discrete set D.  ... 
arXiv:1106.1196v1 fatcat:ik5lb72e6nhgtm3r746en3m7nq

O-MINIMALISM

HANS SCHOUTENS
2014 Journal of Symbolic Logic (JSL)  
As an application, we study certain analytic subsets, called Taylor sets.  ...  , provided one replaces finiteness by discreteness in all of these.  ...  (which in the o-minimal case does yield a finite cell decomposition), we can decompose each X i as a disjoint union of ∅-definable subsets X (e) i consisting of the union of all e-cells in a cell decomposition  ... 
doi:10.1017/jsl.2013.14 fatcat:hcrqbgyihfej3kt424wfrgeyuu

Definable Sets in Ordered Structures. II

Julia F. Knight, Anand Pillay, Charles Steinhorn
1986 Transactions of the American Mathematical Society  
It is simultaneously proved that if M is 0minimal, then every definable set of n-tuples of M has finitely many "definably connected components."  ...  Then any definable X C Mn is a disjoint union of finitely many definably connected definable sets.  ...  Roughly speaking, our proofs of the main theorems will go as follows: We will prove inductively, for each n: (i) any definable set X in Mn is a finite union of cells, (ii) any definable function /: Mn  ... 
doi:10.2307/2000053 fatcat:5juus5s6hnallbby6lqmlavmna

Locally o-minimal structures

Tomohiro KAWAKAMI, Kota TAKEUCHI, Hiroshi TANAKA, Akito TSUBOI
2012 Journal of the Mathematical Society of Japan  
We first give a characterization of the strong local o-minimality. We also investigate locally o-minimal expansions of (R, +, <).  ...  As in the o-minimal setting, we can define cells and cell decompositions of definable sets in the locally o-minimal setting, see [3] .  ...  Recall that M is said to be o-minimal if every definable subset of M is a finite union of points and open intervals in M .  ... 
doi:10.2969/jmsj/06430783 fatcat:hx5zznke65faxpeityc6b6zzry

Definable sets in ordered structures. II

Julia F. Knight, Anand Pillay, Charles Steinhorn
1986 Transactions of the American Mathematical Society  
It is simultaneously proved that if M is 0minimal, then every definable set of n-tuples of M has finitely many "definably connected components."  ...  Then any definable X C Mn is a disjoint union of finitely many definably connected definable sets.  ...  Roughly speaking, our proofs of the main theorems will go as follows: We will prove inductively, for each n: (i) any definable set X in Mn is a finite union of cells, (ii) any definable function /: Mn  ... 
doi:10.1090/s0002-9947-1986-0833698-1 fatcat:3h5h3dhxczcsjofwd3rr5vyuwu

On groups and fields definable in o-minimal structures

Anand Pillay
1988 Journal of Pure and Applied Algebra  
The structure M = (M, <, R" R" .) is o-minimal if every definable set XC M is a finite union of intervals (a, b) and points. Let G be a group definable in M (i.e.  ...  G is a definable subset of M" and the graph of multiplication is also definable).  ...  First, any definable subset of V is a finite union of definable cells, each of which is locally closed (i.e. the intersection of an open set and a closed set) in Mk.  ... 
doi:10.1016/0022-4049(88)90125-9 fatcat:dmn7wheppncj5e33jzjqb4yvgq

Page 458 of American Mathematical Society. Proceedings of the American Mathematical Society Vol. 5, Issue 3 [page]

1954 American Mathematical Society. Proceedings of the American Mathematical Society  
It is easy to see that the closure of S,,—P 1 can be expressed as the union of a finite set oi, - - - , ¢, of elements of =.  ...  An integer k and an (w—1)-cell J* are chosen as in case 2, and sets Sm(€) are defined as in case 2. We also choose 6 as in case 2.  ... 

Expansions of o-minimal structures by sparse sets

Harvey Friedman, Chris Miller
2001 Fundamenta Mathematicae  
The same holds true with "nowhere dense" replaced by any of "null" (in the sense of Lebesgue), "countable", "a finite union of discrete sets", or "discrete".  ...  The same holds true with "nowhere dense" replaced (uniformly) by any of "null" (in the sense of Lebesgue), "countable", "a finite union of discrete sets", or "discrete".  ...  By Theorem B, every subset of R definable in (R, +, ·, Fib) # , as well as every subset of R definable in (R, +, ·, ϕ Z ) # , is the union of an open set and finitely many discrete sets.  ... 
doi:10.4064/fm167-1-4 fatcat:52h3xzzfzjbtro5vwwgi4qrnoa
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