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.

Table of contents |

2 Relation to separation axioms and separated spaces 3 Relation to connected spaces 4 Relation to topologically distinguishable points |

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 intervalss [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).

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 T_{2} 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 *T _{2} 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.

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.

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.