Lusin space
Encyclopedia
In mathematics, a Lusin space or Luzin space, named for N. N. Luzin, may mean:
- A Lusin space in measure theory and probabilityProbabilityProbability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
, the image of a Polish spacePolish spaceIn 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...
under a bijective continuous map. (Some authors add the condition that it should be metrizable.) - A Luzin spaceLuzin setIn real analysis and descriptive set theory, a Luzin set , named for N. N. Luzin, is an uncountable subset A of the reals such that every uncountable subset of A is nonmeager; that is, of second Baire category. Equivalently, A is an uncountable set of reals which meets every first category set in...
in general topologyGeneral topologyIn mathematics, general topology or point-set topology is the branch of topology which studies properties of topological spaces and structures defined on them...
, an uncountable topological T1-space without isolated points in which every nowhere-dense subset is countable. - A Luzin setLuzin setIn real analysis and descriptive set theory, a Luzin set , named for N. N. Luzin, is an uncountable subset A of the reals such that every uncountable subset of A is nonmeager; that is, of second Baire category. Equivalently, A is an uncountable set of reals which meets every first category set in...
in set theorySet theorySet theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
, an uncountable subset of the reals that has at most countable intersection with every nowhere-dense subset of the reals.