The following two assertions are equivalent for an o-minimal expansion of ordered group $\mathcal M=(M,<,+,0,\ldots)$. There exists a definable bijection between bounded interval and unbounded interval. Any continuous function $f:A \rightarrow M$ defined on closed subset $M^n$ has extension $F:M^n M$.

Let R be an o-minimal expansion of a real closed field. Given definable continuous functions f : U → R and : U → (0,+∞), where U is an open subset of Rn, we construct a definable Cm-function g : U → R with |g(x)− f(x)| < (x) for all x ∈ U . Moreover, we show that if f is uniformly continuous, then g can also chosen to be uniformly continuous.

Let R be an expansion of a dense linear order (R,<) without endpoints having the intermediate value property, that is, for all a, b ∈ R, every continuous (parametrically) definable function f : [a, b] → R takes on all values in R between f(a) and f(b). Every expansion of the real line (R, <), as well as every o-minimal expansion of (R,<), has the intermediate value property. Conversely, some ni...

The question as to whether the propositional logic of Heyting, which was a formalization of Brouwer's intuitionistic logic, is finitely many valued or not, was open for a while (the question was asked by Hahn). Kurt Gödel (1932) introduced an infinite decreasing chain of intermediate logics, which are known nowadays as Gödel logics, for showing that the intuitionistic logic is not finitely (man...

We study definably complete locally o-minimal expansions of ordered groups in this paper. A definable continuous function defined on a closed, bounded and set behave like compact set. demonstrate uniform continuity Arzela-Ascoli-type theorem. propose notion special submanifolds with tubular neighborhoods show that any is decomposed into finitely many neighborhoods.

Let (U ;D) be a Gr-covering approximation space (U ; C) with covering lower approximation operator D and covering upper approximation operator D. For a subset X of U , this paper investigates the following three conditions: (1)X is a definable subset of (U ;D); (2) X is an inner definable subset of (U ;D); (3) X is an outer definable subset of (U ;D). It is proved that if one of the above three...

We observe simple links between preorders, semi-groups, rings and categories (and between equivalence relations, groups, fields and groupoids), which are type-definable in an arbitrary structure, and apply these observations to small structures. Recall that a structure is small if it has countably many pure n-types for each integer n. A ∅-type-definable group of finite arity in a small structur...