general topology In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometri ...

, a subset of a

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...

is perfect if it is

closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...

and has no

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 equiva ...

s. Equivalently: the set

S

is perfect if

S=S'

, where

S'

denotes the set of all Limit point, limit points of

S

, also known as the derived set of

S

. In a perfect set, every point can be approximated arbitrarily well by other points from the set: given any point of

S

and any

neighborhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...

of the point, there is another point of

S

that lies within the neighborhood. Furthermore, any point of the space that can be so approximated by points of

S

belongs to

S

. Note that the term ''perfect space'' is also used, incompatibly, to refer to other properties of a topological space, such as being a G_δ space. As another possible source of confusion, also note that having the perfect set property is not the same as being a perfect set.

Examples

Examples of perfect subsets of the

real line In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a po ...

\mathbb

are the

empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in othe ...

, all closed intervals, the real line itself, and the

Cantor set In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883. T ...

. The latter is noteworthy in that it is

totally disconnected In topology and related branches of mathematics, a totally disconnected space is a topological space that has only singletons as connected subsets. In every topological space, the singletons (and, when it is considered connected, the empty set) ...

. Whether a set is perfect or not (and whether it is closed or not) depends on the surrounding space. For instance, the set

cap \Q

is perfect as a subset of the space

\Q

but not perfect as a subset of the space

\mathbb

Connection with other topological properties

Every topological space can be written in a unique way as the disjoint union of a perfect set and a scattered set.

Cantor A cantor or chanter is a person who leads people in singing or sometimes in prayer. In formal Jewish worship, a cantor is a person who sings solo verses or passages to which the choir or congregation responds. In Judaism, a cantor sings and lead ...

proved that every closed subset of the real line can be uniquely written as the disjoint union of a perfect set and a

countable set 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 numb ...

. This is also true more generally for all closed subsets of Polish spaces, in which case the theorem is known as the Cantor–Bendixson theorem. Cantor also showed that every non-empty perfect subset of the real line has

cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...

2^

, the

cardinality of the continuum In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers \mathbb R, sometimes called the continuum. It is an infinite cardinal number and is denoted by \mathfrak c (lowercase fraktur "c") or , \math ...

. These results are extended in

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 ...

as follows: * If ''X'' is a

complete metric space In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in . Intuitively, a space is complete if there are no "points missing" from it (inside or at the bou ...

with no isolated points, then the

Cantor space In mathematics, a Cantor space, named for Georg Cantor, is a topological abstraction of the classical Cantor set: a topological space is a Cantor space if it is homeomorphic to the Cantor set. In set theory, the topological space 2ω is called "the ...

2^ω can be continuously embedded into ''X''. Thus ''X'' has cardinality at least

2^

. If ''X'' is a separable, complete metric space with no isolated points, the cardinality of ''X'' is exactly

2^

. * If ''X'' is a

locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...

Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the ma ...

with no isolated points, there is an

injective function In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositi ...

(not necessarily continuous) from Cantor space to ''X'', and so ''X'' has cardinality at least

2^

Notes

References

* Engelking, Ryszard, ''General Topology'', Heldermann Verlag Berlin, 1989. * * * {{Citation , editor1-last=Pearl , editor1-first=Elliott , title=Open problems in topology. II , publisher=

Elsevier Elsevier () is a Dutch academic publishing company specializing in scientific, technical, and medical content. Its products include journals such as '' The Lancet'', '' Cell'', the ScienceDirect collection of electronic journals, '' Trends'', ...

, isbn=978-0-444-52208-5, mr=2367385 , year=2007 Topology Properties of topological spaces

Examples

Connection with other topological properties

See also

Notes

References