Editing Separated sets (section)

==Definitions==

There are various ways in which two subsets <math>A</math> and <math>B</math> of a topological space <math>X</math> can be considered to be separated.  A most basic way in which two sets can be separated is if they are '''[[disjoint sets|disjoint]]''', that is, if their [[intersection (set theory)|intersection]] is the [[empty set]]. This property has nothing to do with topology as such, but only [[naive set theory|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 <math>A</math> and <math>B</math> are '''{{visible anchor|separated}}''' in <math>X</math> if each is disjoint from the other's [[closure (topology)|closure]]:

<math display=block>A \cap \bar{B} = \varnothing = \bar{A} \cap B.</math>

This property is known as the {{em|Hausdorff−Lennes Separation Condition}}.<ref>{{harvnb|Pervin|1964|loc=p. 51}}</ref> 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 [[interval (mathematics)|interval]]s <math>[0, 1)</math> and <math>(1, 2]</math> are separated in the [[real line]] <math>\Reals,</math> even though the point 1 belongs to both of their closures. A more general example is that in any [[metric space]], two [[open balls]] <math>B_r(p) = \{x \in X : d(p, x) < r\}</math> and <math>B_s(q) = \{x \in X : d(q, x) < s\}</math> are separated whenever <math>d(p, q) \geq r + s.</math> The property of being separated can also be expressed in terms of [[Derived set (mathematics)|derived set]] (indicated by the prime symbol): <math>A</math> and <math>B</math> are separated when they are disjoint and each is disjoint from the other's derived set, that is, <math display=inline>A' \cap B = \varnothing = B' \cap A.</math> (As in the case of the first version of the definition, the derived sets <math>A'</math> and <math>B'</math> are not required to be disjoint from each other.)

The sets <math>A</math> and <math>B</math> are '''{{visible anchor|separated by neighbourhoods}}''' if there are [[neighbourhood (topology)|neighbourhoods]] <math>U</math> of <math>A</math> and <math>V</math> of <math>B</math> such that <math>U</math> and <math>V</math> are disjoint. (Sometimes you will see the requirement that <math>U</math> and <math>V</math> be ''[[Open (topology)|open]]'' neighbourhoods, but this makes no difference in the end.) For the example of <math>A = [0, 1)</math> and <math>B = (1, 2],</math> you could take <math>U = (-1, 1)</math> and <math>V = (1, 3).</math> Note that if any two sets are separated by neighbourhoods, then certainly they are separated. If <math>A</math> and <math>B</math> are open and disjoint, then they must be separated by neighbourhoods; just take <math>U = A</math> and <math>V = B.</math> For this reason, separatedness is often used with closed sets (as in the [[normal separation axiom]]).

The sets <math>A</math> and <math>B</math> are '''{{visible anchor|separated by closed neighbourhoods}}''' if there is a [[closed (topology)|closed]] neighbourhood <math>U</math> of <math>A</math> and a closed neighbourhood <math>V</math> of <math>B</math> such that <math>U</math> and <math>V</math> are disjoint. Our examples, <math>[0, 1)</math> and <math>(1, 2],</math> are {{em|not}} separated by closed neighbourhoods. You could make either <math>U</math> or <math>V</math>  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|separated by neighbourhoods]].

The sets <math>A</math> and <math>B</math> are '''{{visible anchor|separated by a continuous function}}''' if there exists a [[continuous function]] <math>f : X \to \Reals</math> from the space <math>X</math> to the real line <math>\Reals</math> such that <math>A \subseteq f^{-1}(0)</math> and <math>B \subseteq f^{-1}(1)</math>, that is, members of <math>A</math> map to 0 and members of <math>B</math> map to 1. (Sometimes the [[unit interval]] <math>[0, 1]</math> is used in place of <math>\Reals</math> in this definition, but this makes no difference.) In our example, <math>[0, 1)</math> and <math>(1, 2]</math> are not separated by a function, because there is no way to continuously define <math>f</math> at the point 1.<ref>{{cite book|title = Topology | last = Munkres |first = James R. | author-link = James Munkres | page = 211 | edition = 2 | year = 2000 | publisher = Prentice Hall | isbn = 0-13-181629-2}}</ref> If two sets are separated by a continuous function, then they are also [[#separated by closed neighbourhoods|separated by closed neighbourhoods]]; the neighbourhoods can be given in terms of the [[preimage]] of <math>f</math> as <math>U = f^{-1}[-c, c]</math> and <math>V = f^{-1}[1 - c, 1 + c],</math> where <math>c</math> is any [[positive number|positive real number]] less than <math>1/2.</math>

The sets <math>A</math> and <math>B</math> are '''{{visible anchor|precisely separated by a continuous function}}''' if there exists a continuous function <math>f : X \to \Reals</math> such that <math>A = f^{-1}(0)</math> and <math>B = f^{-1}(1).</math> (Again, you may also see the unit interval in place of <math>\Reals,</math> and again it makes no difference.) Note that if any two sets are precisely separated by a function, then they are [[#separated by a continuous function|separated by a function]]. Since <math>\{0\}</math> and <math>\{1\}</math> are closed in <math>\Reals,</math> 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).