Tag: sparse

Uniform finiteness for sparse subsets of the plane

PatrickO-minimal structures / Teaching2 Comments

Let $\M$ be an o-minimal expansion of a dense linear order $(M,\lt)$, and let $S \subseteq M^2$. Our goal is to show that if $S$ is definable and sparse, then $S$ satisfies uniform finitess. For $z \in S$, we say that $\Pi_1\rest{S}$ is a homeomorphism at $z$ if there exists an open box $B \subseteq…

Sparse subsets of the plane

PatrickO-minimal structures / Teaching2 Comments

Let ${\cal M}$ be an o-minimal expansion of a dense linear order $(M,\lt)$, and let $S \subseteq M^2$. $S$ is sparse if $S$ has empty interior. Lemma Assume $S$ is definable. The following are equivalent: $S$ is sparse; the set $S’$ of all $x \in M$ such that $S_x$ is infinite is finite; $S$ is…

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