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In 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 equi ...
s in which every
nowhere-dense In mathematics, a subset of a topological space is called nowhere dense or rare if its closure has empty interior. In a very loose sense, it is a set whose elements are not tightly clustered (as defined by the topology on the space) anywhere. F ...
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 number ...
. 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 ...
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 consi ...
and the negation of 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 ...
, there are no Hausdorff Luzin spaces.


In real analysis

In
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 con ...
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 oth ...
, 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 call ...
; that is, of second Baire category. 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 ''A'' is a subset of a set ''B'' if all 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 unequal, then ''A'' is a proper subset o ...
. 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 ...
, 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 ...
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 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 ...
, every uncountable subset of which has positive outer measure. These sets are called Sierpiński sets, 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 t ...
. Sierpiński sets are weakly Luzin sets but are not Luzin sets.


Example of a Luzin set

Choose a collection of 20 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 subgrou ...
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