Separated sets
Encyclopedia
In topology
and related branches of mathematics
, separated sets are pairs of subset
s of a given topological space
that are related to each other in a certain way.
The notion of when two sets are separated or not is important both to the notion of connected space
s (and their connected components) as well as to the separation axiom
s for topological spaces.
Separated sets should not be confused with separated spaces (defined below), which are somewhat related but different.
Separable spaces are again a completely different topological concept.
As an example, we will define the T2 axiom, which is the condition imposed on separated spaces.
Specifically, a topological space is separated if, given any two distinct points x and y, the singleton sets {x} and {y} are separated by neighbourhoods.
Separated spaces are also called Hausdorff spaces or T2 spaces.
Further discussion of separated spaces may be found in the article Hausdorff space
.
General discussion of the various separation axioms is in the article Separation axiom
.
.
This is certainly true if A is either the empty set or the entire space X, but there may be other possibilities.
A topological space X is connected if these are the only two possibilities.
Conversely, if a nonempty subset A is separated from its own complement, and if the only subset
of A to share this property is the empty set, then A is an open-connected component of X.
(In the degenerate case where X is itself the empty set
{}, authorities differ on whether {} is connected and whether {} is an open-connected component of itself.)
For more on connected spaces, see Connected space
.
that one point belongs to but the other point does not.
If x and y are topologically distinguishable, then the singleton sets {x} and {y} must be disjoint.
On the other hand, if the singletons {x} and {y} are separated, then the points x and y must be topologically distinguishable.
Thus for singletons, topological distinguishability is a condition in between disjointness and separatedness.
For more about topologically distinguishable points, see Topological distinguishability.
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
and related branches of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, separated sets are pairs of subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
s of a given topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
that are related to each other in a certain way.
The notion of when two sets are separated or not is important both to the notion of connected space
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
s (and their connected components) as well as to the separation axiom
Separation axiom
In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. Some of these restrictions are given by the separation axioms...
s for topological spaces.
Separated sets should not be confused with separated spaces (defined below), which are somewhat related but different.
Separable spaces are again a completely different topological concept.
Definitions
There are various ways in which two subsets of a topological space X can be considered to be separated.- A and B are disjoint if their intersectionIntersection (set theory)In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....
is the empty setEmpty setIn mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
. This property has nothing to do with topology as such, but only set theoryNaive set theoryNaive set theory is one of several theories of sets used in the discussion of the foundations of mathematics. The informal content of this naive set theory supports both the aspects of mathematical sets familiar in discrete mathematics , and the everyday usage of set theory concepts in most...
; we include it here because it is the weakest in the sequence of different notions. For more on disjointness in general, see: disjoint sets.
- A and B are separated in X if each is disjoint from the other's closureClosure (topology)In mathematics, the closure of a subset S in a topological space consists of all points in S plus the limit points of S. Intuitively, these are all the points that are "near" S. A point which is in the closure of S is a point of closure of S...
. The closures themselves do not have to be disjoint from each other; for example, the intervalInterval (mathematics)In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...
s [0,1) and (1,2] are separated in the real lineReal lineIn mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...
R, even though the point 1 belongs to both of their closures. More generally in any metric spaceMetric spaceIn mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
, two open balls Br(x1) = {y:d(x1,y)<r} and Bs(x2) = {y:d(x2,y)<s} are separated whenever d(x1,x2) ≥ r+s. Note that any two separated sets automatically must be disjoint.
- A and B are separated by neighbourhoods if there are neighbourhoods U of A and V of B such that U and V are disjoint. (Sometimes you will see the requirement that U and V be open neighbourhoods, but this makes no difference in the end.) For the example of A = [0,1) and B = (1,2], you could take U = (-1,1) and V = (1,3). Note that if any two sets are separated by neighbourhoods, then certainly they are separated. If A and B are open and disjoint, then they must be separated by neighbourhoods; just take U := A and V := B. For this reason, separatedness is often used with closed sets (as in the normal separation axiom).
- A and B are separated by closed neighbourhoods if there is a closed neighbourhood U of A and a closed neighbourhood V of B such that U and V are disjoint. Our examples, [0,1) and (1,2], are not separated by closed neighbourhoods. You could make either U or V closed by including the point 1 in it, but you cannot make them both closed while keeping them disjoint. Note that if any two sets are separated by closed neighbourhoods, then certainly they are separated by neighbourhoods.
- A and B are separated by a function if there exists a continuous functionContinuous functionIn mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
f from the space X to the real line R such that f(A) = {0} and f(B) = {1}. (Sometimes you will see the unit intervalUnit intervalIn mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1...
[0,1] used in place of R in this definition, but it makes no difference in the end.) In our example, [0,1) and (1,2] are not separated by a function, because there is no way to continuously define f at the point 1. Note that if any two sets are separated by a function, then they are also separated by closed neighbourhoods; the neighbourhoods can be given in terms of the preimage of f as U := f-1[-e,e] and V := f-1[1-e,1+e], as long as e is a positive real number less than 1/2.
- A and B are precisely separated by a function if there exists a continuous function f from X to R such that f -1(0) = A and f -1(1) = B. (Again, you may also see the unit interval in place of R, and again it makes no difference.) Note that if any two sets are precisely separated by a function, then certainly they are separated by a function. Since {0} and {1} are closed in R, only closed sets are capable of being precisely separated by a function; but just because two sets are closed and separated by a function does not mean that they are automatically precisely separated by a function (even a different function).
Relation to separation axioms and separated spaces
The separation axioms are various conditions that are sometimes imposed upon topological spaces which can be described in terms of the various types of separated sets.As an example, we will define the T2 axiom, which is the condition imposed on separated spaces.
Specifically, a topological space is separated if, given any two distinct points x and y, the singleton sets {x} and {y} are separated by neighbourhoods.
Separated spaces are also called Hausdorff spaces or T2 spaces.
Further discussion of separated spaces may be found in the article Hausdorff space
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
.
General discussion of the various separation axioms is in the article Separation axiom
Separation axiom
In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. Some of these restrictions are given by the separation axioms...
.
Relation to connected spaces
Given a topological space X, it is sometimes useful to consider whether it is possible for a subset A to be separated from its complementComplement (set theory)
In set theory, a complement of a set A refers to things not in , A. The relative complement of A with respect to a set B, is the set of elements in B but not in A...
.
This is certainly true if A is either the empty set or the entire space X, but there may be other possibilities.
A topological space X is connected if these are the only two possibilities.
Conversely, if a nonempty subset A is separated from its own complement, and if the only subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
of A to share this property is the empty set, then A is an open-connected component of X.
(In the degenerate case where X is itself the empty set
Empty set
In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
{}, authorities differ on whether {} is connected and whether {} is an open-connected component of itself.)
For more on connected spaces, see Connected space
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
.
Relation to topologically distinguishable points
Given a topological space X, two points x and y are topologically distinguishable if there exists an open setOpen set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
that one point belongs to but the other point does not.
If x and y are topologically distinguishable, then the singleton sets {x} and {y} must be disjoint.
On the other hand, if the singletons {x} and {y} are separated, then the points x and y must be topologically distinguishable.
Thus for singletons, topological distinguishability is a condition in between disjointness and separatedness.
For more about topologically distinguishable points, see Topological distinguishability.
Sources
- Stephen Willard, General Topology, Addison-Wesley, 1970. Reprinted by Dover Publications, New York, 2004. ISBN 0-486-43479-6 (Dover edition).