In topology and related branches of mathematics, separated sets are pairs of subsets of a given topological space that are related to each other in a certain way: roughly speaking, neither overlapping nor touching. The notion of when two sets are separated or not is important both to the notion of connected spaces (and their connected components) as well as to the separation axioms 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 and of a topological space can be considered to be separated. A most basic way in which two sets can be separated is if they are disjoint, that is, if their intersection is the empty set. This property has nothing to do with topology as such, but only set theory. Each of the following properties is stricter than disjointness, incorporating some topological information.
The properties below are presented in increasing order of specificity, each being a stronger notion than the preceding one.
The sets and are separated in if each is disjoint from the other's closure:
This property is known as the Hausdorff−Lennes Separation Condition.[1] Since every set is contained in its closure, two separated sets automatically must be disjoint. The closures themselves do not have to be disjoint from each other; for example, the intervals and are separated in the real line even though the point 1 belongs to both of their closures. A more general example is that in any metric space, two open balls and are separated whenever The property of being separated can also be expressed in terms of derived set (indicated by the prime symbol): and are separated when they are disjoint and each is disjoint from the other's derived set, that is, (As in the case of the first version of the definition, the derived sets and are not required to be disjoint from each other.)
The sets and are separated by neighbourhoods if there are neighbourhoods of and of such that and are disjoint. (Sometimes you will see the requirement that and be open neighbourhoods, but this makes no difference in the end.) For the example of and you could take and Note that if any two sets are separated by neighbourhoods, then certainly they are separated. If and are open and disjoint, then they must be separated by neighbourhoods; just take and For this reason, separatedness is often used with closed sets (as in the normal separation axiom).
The sets and are separated by closed neighbourhoods if there is a closed neighbourhood of and a closed neighbourhood of such that and are disjoint. Our examples, and are not separated by closed neighbourhoods. You could make either or 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.
The sets and are separated by a continuous function if there exists a continuous function from the space to the real line such that and , that is, members of map to 0 and members of map to 1. (Sometimes the unit interval is used in place of in this definition, but this makes no difference.) In our example, and are not separated by a function, because there is no way to continuously define at the point 1.[2] If two sets are separated by a continuous function, then they are also separated by closed neighbourhoods; the neighbourhoods can be given in terms of the preimage of as and where is any positive real number less than
The sets and are precisely separated by a continuous function if there exists a continuous function such that and (Again, you may also see the unit interval in place of and again it makes no difference.) Note that if any two sets are precisely separated by a function, then they are separated by a function. Since and are closed in 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, many of 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 usually called Hausdorff spaces or T2 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 complement. 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.)
Given a topological space X, two points x and y are topologically distinguishable if there exists an open set 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.
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it includes all limiting values of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval [0,1] would be compact. Similarly, the space of rational numbers is not compact, because it has infinitely many "punctures" corresponding to the irrational numbers, and the space of real numbers is not compact either, because it excludes the two limiting values and . However, the extended real number linewould be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other topological spaces.
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 non-empty open subsets. Connectedness is one of the principal topological properties that are used to distinguish topological spaces.
In topology and related branches of mathematics, a Hausdorff space ( HOWSS-dorf, HOWZ-dorf), T2 space or separated space, is a topological space where distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" (T2) is the most frequently used and discussed. It implies the uniqueness of limits of sequences, nets, and filters.
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seeming, topology called the box topology, which can also be given to a product space and which agrees with the product topology when the product is over only finitely many spaces. However, the product topology is "correct" in that it makes the product space a categorical product of its factors, whereas the box topology is too fine; in that sense the product topology is the natural topology on the Cartesian product.
In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence.
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 points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through open sets, which is easier than the others to manipulate.
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. For a list of terms specific to algebraic topology, see Glossary of algebraic topology.
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.
A subset of a topological space is called a regular open set if it is equal to the interior of its closure; expressed symbolically, if or, equivalently, if where and denote, respectively, the interior, closure and boundary of
In topology and related branches of mathematics, a normal space is a topological space X that satisfies Axiom T4: every two disjoint closed sets of X have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space. These conditions are examples of separation axioms and their further strengthenings define completely normal Hausdorff spaces, or T5 spaces, and perfectly normal Hausdorff spaces, or T6 spaces.
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 every point has a compact neighborhood.
In topology, Urysohn's lemma is a lemma that states that a topological space is normal if and only if any two disjoint closed subsets can be separated by a continuous function.
In topology and related fields of mathematics, a topological space X is called a regular space if every closed subset C of X and a point p not contained in C have non-overlapping open neighborhoods. Thus p and C can be separated by neighborhoods. This condition is known as Axiom T3. The term "T3 space" usually means "a regular Hausdorff space". These conditions are examples of separation axioms.
In the mathematical field of topology, the Alexandroff extension is a way to extend a noncompact topological space by adjoining a single point in such a way that the resulting space is compact. It is named after the Russian mathematician Pavel Alexandroff. More precisely, let X be a topological space. Then the Alexandroff extension of X is a certain compact space X* together with an open embedding c : X → X* such that the complement of X in X* consists of a single point, typically denoted ∞. The map c is a Hausdorff compactification if and only if X is a locally compact, noncompact Hausdorff space. For such spaces the Alexandroff extension is called the one-point compactification or Alexandroff compactification. The advantages of the Alexandroff compactification lie in its simple, often geometrically meaningful structure and the fact that it is in a precise sense minimal among all compactifications; the disadvantage lies in the fact that it only gives a Hausdorff compactification on the class of locally compact, noncompact Hausdorff spaces, unlike the Stone–Čech compactification which exists for any topological space.
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, geometric topology, and algebraic topology.
In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient topology, that is, with the finest topology that makes continuous the canonical projection map. In other words, a subset of a quotient space is open if and only if its preimage under the canonical projection map is open in the original topological space.
In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological property is a proper class of topological spaces which is closed under homeomorphisms. That is, a property of spaces is a topological property if whenever a space X possesses that property every space homeomorphic to X possesses that property. Informally, a topological property is a property of the space that can be expressed using open sets.
In mathematics, a partition topology is a topology that can be induced on any set by partitioning into disjoint subsets these subsets form the basis for the topology. There are two important examples which have their own names:
In topology and other branches of mathematics, a topological space X is locally connected if every point admits a neighbourhood basis consisting of open connected sets.
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. These are sometimes called Tychonoff separation axioms, after Andrey Tychonoff.
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