mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a Luzin space (or Lusin space), named for N. N. Luzin, is an

uncountable In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal numb ...

topological T₁ space without

isolated point ] In mathematics, a point ''x'' is called an isolated point of a subset ''S'' (in a topological space ''X'') if ''x'' is an element of ''S'' and there exists a neighborhood of ''x'' which does not contain any other points of ''S''. This is equival ...

s in which every nowhere-dense subset is

countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...

. There are many minor variations of this definition in use: the T₁ condition can be replaced by T₂ or T₃, and some authors allow a countable or even arbitrary number of isolated points. The existence of a Luzin space is independent of the axioms of ZFC. showed that the

continuum hypothesis In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that or equivalently, that In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to ...

implies that a Luzin space exists. showed that assuming

Martin's axiom In the mathematical field of set theory, Martin's axiom, introduced by Donald A. Martin and Robert M. Solovay, is a statement that is independent of the usual axioms of ZFC set theory. It is implied by the continuum hypothesis, but it is consist ...

and the negation of the

, there are no Hausdorff Luzin spaces.

In real analysis

real analysis In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include converg ...

and

descriptive set theory In mathematical logic, descriptive set theory (DST) is the study of certain classes of "well-behaved" subsets of the real line and other Polish spaces. As well as being one of the primary areas of research in set theory, it has applications to ot ...

, a Luzin set (or Lusin set), is defined as an uncountable subset of the reals such that every uncountable subset of is

nonmeager In the mathematical field of general topology, a meagre set (also called a meager set or a set of first category) is a subset of a topological space that is small or negligible in a precise sense detailed below. A set that is not meagre is calle ...

; that is, of second

Baire category In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior. According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are ex ...

. Equivalently, is an uncountable set of reals that meets every first category set in only countably many points. Luzin proved that, if the continuum hypothesis holds, then every nonmeager set has a Luzin

subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...

. Obvious properties of a Luzin set are that it must be nonmeager (otherwise the set itself is an uncountable meager subset) and of

measure zero In mathematical analysis, a null set N \subset \mathbb is a measurable set that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length. The notion of null s ...

, because every set of positive measure contains a meager set that also has positive measure, and is therefore uncountable. A weakly Luzin set is an uncountable subset of a real

vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...

such that for any uncountable subset the set of directions between different elements of the subset is dense in the sphere of directions. The measure-category duality provides a

measure Measure may refer to: * Measurement, the assignment of a number to a characteristic of an object or event Law * Ballot measure, proposed legislation in the United States * Church of England Measure, legislation of the Church of England * Mea ...

analogue of Luzin sets – sets of positive

outer measure In the mathematical field of measure theory, an outer measure or exterior measure is a function defined on all subsets of a given set with values in the extended real numbers satisfying some additional technical conditions. The theory of outer mea ...

, every uncountable subset of which has positive outer measure. These sets are called

Sierpiński set In mathematics, a Sierpiński set is an uncountable subset of a real vector space whose intersection with every measure-zero set is countable. The existence of Sierpiński sets is independent of the axioms of ZFC. showed that they exist if the c ...

s, after

Wacław Sierpiński Wacław Franciszek Sierpiński (; 14 March 1882 – 21 October 1969) was a Polish mathematician. He was known for contributions to set theory (research on the axiom of choice and the continuum hypothesis), number theory, theory of functions, and to ...

. Sierpiński sets are weakly Luzin sets but are not Luzin sets.

Example of a Luzin set

Choose a collection of 2^ℵ₀ meager subsets of R such that every meager subset is contained in one of them. By the continuum hypothesis, it is possible to enumerate them as ''S''_''α'' for countable ordinals ''α''. For each countable ordinal ''β'' choose a real number ''x''_''β'' that is not in any of the sets ''S''_''α'' for ''α''<''β'', which is possible as the union of these sets is meager so is not the whole of R. Then the uncountable set ''X'' of all these real numbers ''x''_''β'' has only a countable number of elements in each set ''S''_''α'', so is a Luzin set. More complicated variations of this construction produce examples of Luzin sets that are

subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...

s, subfields or real-closed subfields of the real numbers.

References

* Paper mentioning Luzin spaces * * * *{{citation, first=John C., last= Oxtoby , authorlink = John C. Oxtoby, title=Measure and category: a survey of the analogies between topological and measure spaces , publisher=Springer-Verlag , location=Berlin , year=1980 , isbn=0-387-90508-1 Properties of topological spaces Descriptive set theory