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In mathematics, more specifically in
point-set 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, geomet ...
, the derived set of a subset $S$ 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 poi ...
is the set of all
limit point In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also cont ...
s of $S.$ It is usually denoted by $S\text{'}.$ The concept was first introduced by
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance o ...
in 1872 and he developed
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concer ...
in large part to study derived sets on 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 ...
.

# Examples

If $\mathbb$ is endowed with its usual Euclidean topology then the derived set of the half-open interval _is_the_closed_interval_$\left[0,1$_ Consider_$\mathbb$_with_the_Topology_(structure).html" "title=",1.html" ;"title=", 1) is the closed interval $\left[0,1">, 1\right)$ is the closed interval $\left[0,1$ Consider $\mathbb$ with the Topology (structure)">topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
(open sets) consisting of the empty set and any subset of $\mathbb$ that contains 1. The derived set of $A := \$ is $A\text{'} = \mathbb \setminus \.$

# Properties

If $A$ and $B$ are subsets of the topological space $\left\left(X, \mathcal\right\right),$ then the derived set has the following properties: * $\varnothing\text{'} = \varnothing$ * $a \in A\text{'} \implies a \in \left(A \setminus \\right)\text{'}$ * $\left(A \cup B\right)\text{'} = A\text{'} \cup B\text{'}$ * $A \subseteq B \implies A\text{'} \subseteq B\text{'}$ A subset $S$ of a topological space is closed precisely when $S\text{'} \subseteq S,$ that is, when $S$ contains all its limit points. For any subset $S,$ the set $S \cup S\text{'}$ is closed and is the closure of $S$ (i.e. the set $\overline$). The derived set of a subset of a space $X$ need not be closed in general. For example, if $X = \$ with the
trivial topology In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the consequ ...
, the set $S = \$ has derived set $S\text{'} = \,$ which is not closed in $X.$ But the derived set of a closed set is always closed. (''Proof:'' Assuming $S$ is a closed subset of $X,$ which shows that $S\text{'} \subseteq S,$ take the derived set on both sides to get $S\text{'}\text{'} \subseteq S\text{'},$ i.e., $S\text{'}$ is closed in $X.$) In addition, if $X$ is a T1 space, the derived set of every subset of $X$ is closed in $X.$ Two subsets $S$ and $T$ are separated precisely when they are disjoint and each is disjoint from the other's derived set (though the derived sets don't need to be disjoint from each other). This condition is often, using closures, written as :$\left\left( S \cap \bar \right\right) \cup \left\left( \bar \cap T \right\right) = \varnothing,$ and is known as the . A
bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
between two topological spaces is a
homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomo ...
if and only if the derived set of the image (in the second space) of any subset of the first space is the image of the derived set of that subset. A space is a T1 space if every subset consisting of a single point is closed. In a T1 space, the derived set of a set consisting of a single element is empty (Example 2 above is not a T1 space). It follows that in T1 spaces, the derived set of any finite set is empty and furthermore, :$\left\left( S - \ \right\right)\text{'} = S\text{'} = \left\left( S \cup \ \right\right)\text{'},$ for any subset $S$ and any point $p$ of the space. In other words, the derived set is not changed by adding to or removing from the given set a finite number of points. It can also be shown that in a T1 space, $\left\left( S\text{'} \right\right)\text{'} \subseteq S\text{'}$ for any subset $S.$ A set $S$ with $S \subseteq S\text{'}$ is called dense-in-itself and can contain 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. A set $S$ with $S = S\text{'}$ is called perfect set, perfect. Equivalently, a perfect set is a closed dense-in-itself set, or, put another way, a closed set with no isolated points. Perfect sets are particularly important in applications of the
Baire category theorem The Baire category theorem (BCT) is an important result in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space (a topological space such that t ...
. The Cantor–Bendixson theorem states that any
Polish space In 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 b ...
can be written as the union of a countable set and a perfect set. Because any Gδ subset of a Polish space is again a Polish space, the theorem also shows that any Gδ subset of a Polish space is the union of a countable set and a set that is perfect with respect to the
induced topology In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced ...
.

# Topology in terms of derived sets

Because homeomorphisms can be described entirely in terms of derived sets, derived sets have been used as the primitive notion in
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
. A set of points $X$ can be equipped with an operator $S \mapsto S^$ mapping subsets of $X$ to subsets of $X,$ such that for any set $S$ and any point $a$: # $\varnothing^* = \varnothing$ # $S^ \subseteq S^*\cup S$ # $a \in S^*$ implies $a \in \left(S \setminus \\right)^*$ # $\left(S \cup T\right)^* \subseteq S^* \cup T^*$ # $S \subseteq T$ implies $S^* \subseteq T^*.$ Calling a set $S$ if $S^ \subseteq S$ will define a topology on the space in which $S \mapsto X^*$ is the derived set operator, that is, $S^ = S\text{'}.$

# Cantor–Bendixson rank

For
ordinal number In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the least ...
s $\alpha,$ the $\alpha$-th Cantor–Bendixson derivative of a topological space is defined by repeatedly applying the derived set operation using
transfinite recursion Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers. Its correctness is a theorem of ZFC. Induction by cases Let P(\alpha) be a property defined for ...
as follows: *$\displaystyle X^0 = X$ *$\displaystyle X^ = \left\left( X^ \right\right)\text{'}$ *$\displaystyle X^ = \bigcap_ X^$ for
limit ordinal In set theory, a limit ordinal is an ordinal number that is neither zero nor a successor ordinal. Alternatively, an ordinal λ is a limit ordinal if there is an ordinal less than λ, and whenever β is an ordinal less than λ, then there exists an ...
s $\lambda.$ The transfinite sequence of Cantor–Bendixson derivatives of $X$ must eventually be constant. The smallest ordinal $\alpha$ such that $X^ = X^$ is called the Cantor–Bendixson rank of $X.$ This investigations into the derivation process was one of the motivations for introducing
ordinal numbers In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the least ...
by Georg Cantor.

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