Separated sets

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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.

Separation axioms
in topological spaces
Kolmogorov classification
T0 (Kolmogorov)
T1 (Fréchet)
T2 (Hausdorff)
T2½(Urysohn)
completely T2 (completely Hausdorff)
T3 (regular Hausdorff)
T(Tychonoff)
T4 (normal Hausdorff)
T5 (completely normal
 Hausdorff)
T6 (perfectly normal
 Hausdorff)

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

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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

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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.

Relation to connected spaces

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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.)

Relation to topologically distinguishable points

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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.

See also

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Citations

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  1. ^ Pervin 1964, p. 51
  2. ^ Munkres, James R. (2000). Topology (2 ed.). Prentice Hall. p. 211. ISBN 0-13-181629-2.

Sources

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