Definability properties

• Borel set
  Borel set
  In mathematics, a Borel set is any set in a topological space that can be formed from open sets through the operations of countable union, countable intersection, and relative complement...
  
   
• Analytic set
  Analytic set
  In descriptive set theory, a subset of a Polish space X is an analytic set if it is a continuous image of a Polish space. These sets were first defined by and his student .- Definition :There are several equivalent definitions of analytic set...
  
   
• C-measurable set 
• Projective set 
• Inductive set
  Inductive set
  In descriptive set theory, an inductive set of real numbers is one that can be defined as the least fixed point of a monotone operation definable by a positive Σ1n formula, for some natural number n, together with a real parameter.The inductive sets form a boldface pointclass; that is, they...
  
   
• Infinity-Borel set
  Infinity-Borel set
  In set theory, a subset of a Polish space X is ∞-Borel if itcan be obtained by starting with the open subsets of X, and transfinitely iterating the operations of complementation and wellordered union...
  
   
• Suslin set
  Suslin set
  The concept of a Suslin set was first used by Mikhail Yakovlevich Suslin when he was researching the properties of projections of Borel sets in \R^2 onto the real axis...
  
   
• Homogeneously Suslin set
  Homogeneously Suslin set
  In descriptive set theory, a set S is said to be homogeneously Suslin if it is the projection of a homogeneous tree. S is said to be \kappa-homogeneously Suslin if it is the projection of a \kappa-homogeneous tree....
  
   
• Weakly homogeneously Suslin set 

Regularity properties

• Property of Baire 
• Lebesgue measurable 
• Universally measurable set
  Universally measurable set
  In mathematics, a subset A of a Polish space X is universally measurable if it is measurable with respect to every complete probability measure on X that measures all Borel subsets of X. In particular, a universally measurable set of reals is necessarily Lebesgue measurable below.Every analytic...
  
   
• Perfect set property
  Perfect set property
  In descriptive set theory, a subset of a Polish space has the perfect set property if it is either countable or has a nonempty perfect subset...
  
   
• Universally Baire set
  Universally Baire set
  In the mathematical field of descriptive set theory, a set of reals is called universally Baire if it has a certain strong regularity property. Universally Baire sets play an important role in Ω-logic, a very strong logical system invented by W...
  
   

Largeness and smallness properties

• Meager set 
• Comeager set - A comeager set is one whose complement is meager.
• Null set
  Null set
  In mathematics, a null set is a set that is negligible in some sense. For different applications, the meaning of "negligible" varies. In measure theory, any set of measure 0 is called a null set...
  
   
• Conull set
  Conull set
  In measure theory, a conull set is a set whose complement is null, i.e., the measure of the complement is zero. For example, the set of irrational numbers is a conull subset of the real line with Lebesgue measure.See also:*Almost everywhere...
  
   
• Dense set
  Dense set
  In topology and related areas of mathematics, a subset A of a topological space X is called dense if any point x in X belongs to A or is a limit point of A...
  
   
• Nowhere dense set
  Nowhere dense set
  In mathematics, a nowhere dense set in a topological space is a set whose closure has empty interior. The order of operations is important. For example, the set of rational numbers, as a subset of R has the property that the closure of the interior is empty, but it is not nowhere dense; in fact it...