In mathematics, more specifically in point-set topology, the derived set of a subset of a topological space is the set of all limit points of It is usually denoted by
The concept was first introduced by Georg Cantor in 1872 and he developed set theory in large part to study derived sets on the real line.
If is endowed with its usual Euclidean topology then the derived set of the half-open interval is the closed interval
Consider with the topology (open sets) consisting of the empty set and any subset of that contains 1. The derived set of is
If and are subsets of the topological space then the derived set has the following properties:
A subset of a topological space is closed precisely when that is, when contains all its limit points. For any subset the set is closed and is the closure of (i.e. the set ).
The derived set of a subset of a space need not be closed in general. For example, if with the trivial topology, the set has derived set which is not closed in But the derived set of a closed set is always closed. (Proof: Assuming is a closed subset of which shows that take the derived set on both sides to get i.e., is closed in ) In addition, if is a T1 space, the derived set of every subset of is closed in
Two subsets and 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
and is known as the Hausdorff-Lennes Separation Condition.
A bijection between two topological spaces is a homeomorphism 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,
for any subset and any point 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, for any subset
A set with is called dense-in-itself and can contain no isolated points. A set with is called 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 Cantor–Bendixson theorem states that any Polish space 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.
Because homeomorphisms can be described entirely in terms of derived sets, derived sets have been used as the primitive notion in topology. A set of points can be equipped with an operator mapping subsets of to subsets of such that for any set and any point :
Calling a set closed if will define a topology on the space in which is the derived set operator, that is,
For ordinal numbers the -th Cantor–Bendixson derivative of a topological space is defined by repeatedly applying the derived set operation using transfinite recursion as follows:
The transfinite sequence of Cantor–Bendixson derivatives of must eventually be constant. The smallest ordinal such that is called the Cantor–Bendixson rank of
In mathematics, a filter or order filter is a special subset of a partially ordered set. Filters appear in order and lattice theory, but can also be found in topology, from which they originate. The dual notion of a filter is an order ideal.
In mathematical analysis and in probability theory, a σ-algebra on a set X is a collection of subsets of X that includes X itself, is closed under complement, and is closed under countable unions.
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 of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.
This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also fundamental to algebraic topology, differential topology and geometric topology.
In commutative algebra the prime spectrum of a ring like R is the set of all prime ideals of R which is usually denoted by , in algebraic geometry it is simultaneously a topological space equipped with the sheaf of rings .
In mathematics, open sets are a generalization of open intervals in the real line. In a metric space—that is, when a distance function is defined—open sets are the sets that, with every point P, contain all points that are sufficiently near to P.
In mathematics, the closure of a subset S of points in a topological space consists of all points in S together with all limit points of S. The closure of S may equivalently be defined as the union of S and its boundary, and also as the intersection of all closed sets containing S. Intuitively, the closure can be thought of as all the points that are either in S or "near" S. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to the notion of interior.
In mathematics, a topological vector space is one of the basic structures investigated in functional analysis. A topological vector space is a vector space which is also a topological space, this implies that vector space operations be continuous functions. More specifically, its topological space has a uniform topological structure, allowing a notion of uniform convergence.
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a closed set is a set which is closed under the limit operation. This should not be confused with a closed manifold.
In mathematics, specifically in topology, the interior of a subset S of a topological space X is the union of all subsets of S that are open in X. A point that is in the interior of S is an interior point of S.
In mathematics, a base or basis for the topology τ of a topological space (X, τ) is a family B of open subsets of X such that every open set of the topology is equal to a union of some sub-family of B. For example, the set of all open intervals in the real number line is a basis for the Euclidean topology on because every open interval is an open set, and also every open subset of can be written as a union of some family of open intervals.
In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of not belonging to the interior of An element of the boundary of is called a boundary point of The term boundary operation refers to finding or taking the boundary of a set. Notations used for boundary of a set include and Some authors use the term frontier instead of boundary in an attempt to avoid confusion with a different definition used in algebraic topology and the theory of manifolds. Despite widespread acceptance of the meaning of the terms boundary and frontier, they have sometimes been used to refer to other sets. For example, Metric Spaces by E. T. Copson uses the term boundary to refer to Hausdorff's border, which is defined as the intersection of a set with its boundary. Hausdorff also introduced the term residue, which is defined as the intersection of a set with the closure of the border of its complement.
In mathematics, the lower limit topology or right half-open interval topology is a topology defined on the set of real numbers; it is different from the standard topology on and has a number of interesting properties. It is the topology generated by the basis of all half-open intervals [a,b), where a and b are real numbers.
In mathematics, particularly topology, a cover of a set is a collection of sets whose union includes as a subset. Formally, if is an indexed family of sets then is a cover of if
In topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms that can be used to define a topological structure on a set. They are equivalent to the more commonly used open set definition. They were first formalized by Kazimierz Kuratowski, and the idea was further studied by mathematicians such as Wacław Sierpiński and António Monteiro, among others.
In mathematics, a topological space is usually defined in terms of open sets. However, there are many equivalent characterizations of the category of topological spaces. Each of these definitions provides a new way of thinking about topological concepts, and many of these have led to further lines of inquiry and generalisation.
In topology and related fields of mathematics, a sequential space is a topological space that satisfies a very weak axiom of countability.
In topology, a preclosure operator, or Čech closure operator is a map between subsets of a set, similar to a topological closure operator, except that it is not required to be idempotent. That is, a preclosure operator obeys only three of the four Kuratowski closure axioms.
In mathematics, an adherent point of a subset of a topological space is a point in such that every neighbourhood of contains at least one point of A point is an adherent point for if and only if is in the closure of thus
In mathematics, a polyadic space is a topological space that is the image under a continuous function of a topological power of an Alexandroff one-point compactification of a discrete space.