In the mathematical discipline of descriptive set theory, a scale is a certain kind of object defined on a set of points in some Polish space. This concept is fundamental for studying the structural properties and complexity of sets of points in spaces like the real numbers.
Understanding Scales in Descriptive Set Theory
As stated, a scale is an object defined on a subset of a Polish space. A Polish space is essentially a separable, completely metrizable topological space – familiar examples include the real numbers, Baire space, or Cantor space. The scale is defined on a specific subset of points within such a space.
While the reference defines it broadly as a "certain kind of object," in more technical terms, a scale is typically represented by a sequence of "norms" or "rankings" on the set. These norms assign values to points in the set, providing a way to measure their complexity or "rank" within the set. The sequence of norms must satisfy certain properties related to convergence, ensuring that points "close" to a point in the set (in a specific sense related to the norms) are also in the set.
Scales are powerful tools used to establish regularity properties of sets, especially those belonging to specific complexity classes within the Borel or projective hierarchies.
Key Aspects of a Scale
A scale on a set $A$ within a Polish space $X$ involves:
- Polish Space ($X$): The ambient space where the set $A$ resides.
- Set of Points ($A$): The specific subset of the Polish space on which the scale is defined.
- Sequence of Norms (${\varphin}{n \in \omega}$): A sequence of functions (norms) $\varphi_n: A \to \text{Ordinals}$ (or some other well-ordered set) that satisfy certain lower semi-continuity properties.
The existence of a scale on a set $A$ often implies strong structural properties about $A$.
Importance and Applications
Scales are crucial in descriptive set theory for several reasons:
- Proving Regularity Properties: The existence of a scale on a set is often the key step in proving that the set has desirable properties, such as the Baire property, the perfect set property, or Lebesgue measurability.
- Relating Complexity Classes: Scales are used to understand the structure of sets in the projective hierarchy (sets defined by quantified formulas). For instance, if a set belongs to a certain level of the hierarchy and has a scale, it implies specific structural characteristics.
- Determinacy Theorems: Scales play a vital role in the study of infinite games and determinacy. The existence of scales on sets at certain levels of complexity is equivalent to the determinacy of associated games (like Wadge games), linking set theory, topology, and game theory.
For example, it is a fundamental result that every $\mathbf{\Pi}^1_1$ set (a set at a particular level of complexity) in a Polish space admits a scale. This fact is used to prove that all $\mathbf{\Pi}^1_1$ sets have the Baire property and the perfect set property.
Scales provide a measure of the "definability" or "smoothness" of a set, acting as a bridge between its definition and its topological structure.
