Lusin's separation theorem
Encyclopedia
In descriptive set theory
and mathematical logic
, Lusin's separation theorem states that if A and B are disjoint analytic subsets
of Polish space
, then there is a Borel set
C in the space such that A ⊆ C and B ∩ C = ∅ (Kechris 1995, p. 87). It is named after Nicolas Lusin, who proved it in 1927.
Descriptive set theory
In mathematical logic, descriptive set theory is the study of certain classes of "well-behaved" subsets of the real line and other Polish spaces...
and mathematical logic
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
, Lusin's separation theorem states that if A and B are disjoint analytic subsets
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...
of 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 there is a 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...
C in the space such that A ⊆ C and B ∩ C = ∅ (Kechris 1995, p. 87). It is named after Nicolas Lusin, who proved it in 1927.