Definably Compact Sets in Definable Spaces Relative to O-minimal Structures

Subject Areas : Statistics

Somayyeh Tari ¹ (Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran.)

Received: 2020-02-07 Accepted : 2020-07-12 Published : 2021-09-23

Keywords: ساختار ت-کمینه, فشردگی تعریف‌پذیر, فضاهای تعریف‌‌پذیر وابسته به ساختارهای ت-کمینه,

Abstract :

A structure M is an o-minimal structure if every definable subset X of M is a finite union of intervals and points.Every o-minimal structure is a topological space. A definable set X in M is called a definably compact set if every definable curve in it is completable in X. The definable compactness of a definable set X is equivalent to be closed and bounded.A idea of definable space generalizes theorem of Robson for semialgebraic spaces, A definablespace is not a definable set but it looks like the definable sets. Indeed definable spaces are constructed by the gluing of sets which are not definable but they are relativ to definable sets. In this paper we will recall the concept of definable space and introduce the notion of definable compactness in it. We show that a definable set X in the housdorff definable space is definable compact if and only if it is closed and bounded.

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