The concept of a Suslin set was first used by Mikhail Yakovlevich Suslin

Mikhail Yakovlevich Suslin

Mikhail Yakovlevich Suslin was a Russian mathematician who made major contributions to the fields of general topology and descriptive set theory....

 when he was researching the properties of projections of Borel sets in onto the real axis. Lebesgue believed he had proved that such a projection was also a Borel set, but an error was spotted by Suslin.

Suslin sets have since been used in various areas of mathematics such as potential theory

Potential theory

In mathematics and mathematical physics, potential theory may be defined as the study of harmonic functions.- Definition and comments :The term "potential theory" was coined in 19th-century physics, when it was realized that the fundamental forces of nature could be modeled using potentials which...

, measure theory and the study of fractals.

There is some variation in the ways in which Suslin sets are defined in mathematical literature. In this article we follow what is probably the most common definition.

Definition

In a metric space

Metric space

In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...

 , the Suslin sets are the sets of the form



where is a closed set in X for each finite sequence of positive integers.

Properties of Suslin sets

• Every 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...
  
   is a Suslin set.
• The collection of all Suslin sets is closed under countable unions and countable intersection.
• Not all Suslin sets are Borel sets (in non-trivial metric spaces such as ).
• If A is a subset of a Polish space
  Polish space
  In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named because they were first extensively studied by Polish...
  
   then A is a Suslin set if and only if it is an 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...
  
  .
• In a Polish space, any Suslin set is universally measurable.