We give an example of a dense o-minimal structure in which there is a definable quotient that cannot be eliminated, even after naming parameters. Equivalently, there is an interpretable set which cannot be put in parametrically definable bijection with any definable set. This gives a negative answer to a question of Eleftheriou, Peterzil, and Ramakrishnan. Additionally, we show that interpretable sets in dense o-minimal structures admit definable topologies which are "tame" in several ways: (a) they are Hausdorff, (b) every point has a neighborhood which is definably homeomorphic to a definable set, (c) definable functions are piecewise continuous, (d) definable subsets have finitely many definably connected components, and (e) the frontier of a definable subset has lower dimension than the subset itself. 2010 Mathematical Subject Classification: 03C64 1 Without this proviso, one can produce pathological examples such as the line with doubled origin. Indeed, if X = R × {0, 1} and E is the equivalence relation generated by (x, 0)E(x, 1) for x = 0, then the quotient X/E is the line with doubled origin.
We consider the class of "well-tempered" integer-valued scoring games, which have the property that the parity of the length of the game is independent of the line of play. We consider disjunctive sums of these games, and develop a theory for them analogous to the standard theory of disjunctive sums of normal-play partizan games. We show that the monoid of well-tempered scoring games modulo indistinguishability is cancellative but not a group, and we describe its structure in terms of the group of normal-play partizan games. We also classify Boolean-valued well-tempered scoring games, showing that there are exactly seventy, up to equivalence. arXiv:1112.3610v1 [math.CO] 15 Dec 2011The present paper focuses on a limited class of scoring games, which we call welltempered because they are somewhat analogous to the well-tempered games of Grossman and Siegel [10]. A well-tempered scoring game is one in which the last player to move is predetermined, and does not depend on the course of the game. In an odd-tempered game, the first player to move will also be the last, while in an even-tempered game, the first player will never be the last. These are exactly the scoring games which are trivial when played as partizan games by the normal-play rule. Nevertheless, we show that our class of games, like the class considered by Milnor, is closely related to the standard theory of partizan normal-play games.Our original motivation was the game To Knot or Not to Knot of [13]. Few other games seem to fit into this framework, so we have invented a few. We refer the reader to §8 for examples.The outline of this paper is as follows. In Section 2, we define precisely what we mean by a well-tempered scoring game, how we add them, and what it means for two to be equivalent. In §3, we prove basic facts with these notions, showing that a certain class of special games (those having the property of Milnor and Hanner at even levels) form a wellbehaved abelian group. To a general game G, we can associate two special games G + and G − , which characterize G. In fact, we show that a general game G has a double identity, acting as G + when Left is the final player, and as G − when Right is the final player. In Section 4, we consider variants of disjunctive addition. Specifically, we vary the manner in which the final scores of the summands are combined into a total score. We provide additional motivation for our earlier definition of equivalence, generalize the results of §3, and characterize the pairs (G + , G − ) which can occur. In §5, we show how our class of games is closely related to and characterized by the standard theory of normal-play partizan games. We use this correspondence to give canonical forms for (special) games in §6. In §7 we discuss the theory of {0, 1}-valued games, which is necessary to analyze games like To Knot or Not to Knot. We show that there are seventy {0, 1}-valued games modulo equivalence, but infinitely many three-valued games. We close in § 8 with some examples of well-tempered scoring games, including small...
We construct a nontrivial definable type V field topology on any dp-minimal field [Formula: see text] that is not strongly minimal, and prove that definable subsets of [Formula: see text] have small boundary. Using this topology and its properties, we show that in any dp-minimal field [Formula: see text], dp-rank of definable sets varies definably in families, dp-rank of complete types is characterized in terms of algebraic closure, and [Formula: see text] is finite for all [Formula: see text]. Additionally, by combining the existence of the topology with results of Jahnke, Simon and Walsberg [Dp-minimal valued fields, J. Symbolic Logic 82(1) (2017) 151–165], it follows that dp-minimal fields that are neither algebraically closed nor real closed admit nontrivial definable Henselian valuations. These results are a key stepping stone toward the classification of dp-minimal fields in [Fun with fields, Ph.D. thesis, University of California, Berkeley (2016)].
For an arbitrary field K and K-variety V , we introduce the étale-open topology on the set V (K) of K-points of V . This topology agrees with the Zariski topology, Euclidean topology, and valuation topology when K is separably closed, real closed, p-adically closed, respectively. The étale open topology on A 1 (K) is not discrete if and only if K is large. Topological properties of the étale open topology corresponds to algebraic properties of K.As an application, we show that a large stable field is separably closed.
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