Set Types, Fundamental Properties, and Descriptive Set Theory: A Foundational Mathematical Study

Set theory provides a common language for mathematical structures, while descriptive set theory investigates how subsets of well-behaved topological spaces can be classified by the complexity of their definitions. This paper presents a coherent foundational account of sets, common types of sets, elementary set operations, and principal algebraic properties, before introducing the topological framework of descriptive set theory. Definitions and illustrative examples are given for empty, singleton, finite, infinite, countable, uncountable, equal, equivalent, disjoint, and power sets. Basic identities are stated and proved using element-wise reasoning. The paper then defines topological spaces, Polish spaces, Borel sets, and the Borel hierarchy, and explains the role of descriptive classifications in the study of real numbers and other Polish spaces. The work is expository and foundational: its contribution is the structured presentation connecting elementary set operations with definability in topology, not a claim of a new theorem.