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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. 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 aren't the same thing. And separable spaces are a completely different topological concept.
Definitions
There are various versions of the concept. The terms are defined below, where X is a topological space.
First, two subsets A and B of X are disjoint if their
intersection is the empty set. This property has nothing to do with topology as such, but only set theory; 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 closure. The closures themselves don't have to be
disjoint from each other; for example, the intervals [0,1) and (1,2] are separated in the real
line R, even though the point 1 belongs to both of their closures. Note that any two separated sets automatically must
be disjoint.
A and B are separated by neighbourhoods if there are a neighbourhood U of A and a neighbourhood 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.
A and B are separated by closed neighbourhoods if there are 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 can't 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 function 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 interval [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 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 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.)
For more on connected spaces, see Connected space.
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. General
discussion of the various separation axioms is in the article Separation axiom.
Relation to topologically distinguishable points
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.
For more about topologically distinguishable points, see Topological distinguishability.
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