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 properties below is stricter than disjointness, incorporating some topological information. The properties are presented in increasing order of specificity, each being a stronger notion than the preceding one.
A more restrictive property is that
and
are separated in
if each is disjoint from the other's closure:
![{\displaystyle A\cap {\bar {B}}=\varnothing ={\bar {A}}\cap B.}](//wikimedia.org/api/rest_v1/media/math/render/svg/8f9b81b8432f5a63b6b03e25c3b9cf5a04424379)
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 ![{\displaystyle 1/2.}](//wikimedia.org/api/rest_v1/media/math/render/svg/0e15cf80ad635a94ab6b09444b44368fdf171fd9)
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).