perfect set

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In
general topology In mathematics, general topology is the branch of topology that deals with the basic Set theory, set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topolog ...
, a subset of a
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
is perfect if it is closed and has no
isolated point ] In 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 ...
s. Equivalently: the set $S$ is perfect if $S=S\text{'}$, where $S\text{'}$ 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 British English (BrE, en-GB, or BE) is, according to Oxford Dictionaries, " English as used in Great Britain, as distinct from that used elsewhere". More narrowly, it can refer specifically to the En ...
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 Gδ space. As another possible source of confusion, also note that having the
perfect set property In descriptive set theory, a subset of a Polish space has the perfect set property if it is either countable set, countable or has a nonempty perfect set, perfect subset (Kechris 1995, p. 150). Note that having the perfect set property is ...
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 Line most often refers to: * Line (geometry) In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are One-dim ...
$\mathbb$ are the
empty set In 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 ...
, all closed intervals, the real line itself, and the
Cantor set In mathematics, the Cantor set is a set (mathematics), 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 Ca ...
. The latter is noteworthy in that it is
totally disconnected In topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, defo ...
. Whether a set is perfect or not (and whether it is closed or not) depends on the surrounding space. For instance, the set 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 Judaism, 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 ...
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 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 mo ...
. 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 (mathematics), set is a measure of the number of Element (mathematics), elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19 ...
$2^$, the
cardinality of the continuum In set theory, the cardinality of the continuum is the cardinality or "size" of the Set (mathematics), set of real numbers \mathbb R, sometimes called the Continuum (set theory), continuum. It is an Infinite set, infinite cardinal number and is den ...
. These results are extended in
descriptive set theory In mathematical logic, descriptive set theory (DST) is the study of certain classes of "well-behaved" set (mathematics), 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 a ...
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#In a metric space, Cauchy sequence of points in has a Limit of a sequence, limit that is also in . Intuitively, a space is complete if ther ...
with no isolated points, then the
Cantor space In mathematics, a Cantor space, named for Georg Cantor, is a topology, 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 ca ...
2ω 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 neighbourhood (mathematics), neighbourhoods of each which are disjoint s ...
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 (mathematics), function that maps Distinct (mathematics), distinct elements of its domain to distinct elements; that is, implies . (Equivale ...
(not necessarily continuous) from Cantor space to ''X'', and so ''X'' has cardinality at least $2^$.