Editing Separated sets

{{Short description|Type of relation for subsets of a topological space}}
{{Separation axioms}}
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In [[topology]] and related branches of [[mathematics]], '''separated sets''' are pairs of [[subset]]s 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 space]]s (and their connected components) as well as to the [[separation axiom]]s for topological spaces.

Separated sets should not be confused with [[separated space]]s (defined below), which are somewhat related but different.  [[Separable space]]s are again a completely different topological concept.

==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).

==Relation to separation axioms and separated spaces==
{{main|separation axiom}}

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 T<sub>2</sub> axiom, which is the condition imposed on separated spaces. Specifically, a topological space is ''separated'' if, given any two [[distinct (mathematics)|distinct]] points ''x'' and ''y'', the singleton sets {''x''} and {''y''} are separated by neighbourhoods.

Separated spaces are usually called ''[[Hausdorff space]]s'' or ''T<sub>2</sub> spaces''.

==Relation to connected spaces==
{{main|Connected space}}

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 (set theory)|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]] <math>\emptyset</math>, authorities differ on whether <math>\emptyset</math> is connected and whether <math>\emptyset</math> is an open-connected component of itself.)

==Relation to topologically distinguishable points==
{{main|Topological distinguishability}}

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 set]]s {''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==

* {{annotated link|Hausdorff space}}
* {{annotated link|Locally Hausdorff space}}
* {{annotated link|Separation axiom}}

==Citations==

{{reflist}}

==Sources==

{{refbegin}}
* {{cite book|author-last=Munkres|author-first=James R.|author-link=James Munkres|title=Topology|date=2000|publisher=[[Prentice Hall|Prentice-Hall]]|isbn=0-13-181629-2}} <!-- {{sfn|Munkres|2000|p=}} -->
* {{cite book|author-last=Willard|author-first=Stephen|title=General Topology|date=2004|publisher=[[Addison-Wesley]]|isbn=0-486-43479-6}} <!-- {{sfn|Willard|2004|p=}} -->
* {{citation|last=Pervin|first=William J.|title=Foundations of General Topology|year=1964|publisher=Academic Press}}
{{refend}}

{{Topology}}

[[Category:Separation axioms]]
[[Category:Topology]]