separated sets

There are various ways in which two subsets $A$ and $B$ of a topological space $X$ 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 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 $A$ and $B$ are {{visible anchor|separated}} in $X$ if each is disjoint from the other's closure:

$$A\; \backslash cap\; \backslash bar\{B\}\; =\; \backslash varnothing\; =\; \backslash bar\{A\}\; \backslash cap\; B.$$

This property is known as the {{em|Hausdorff−Lennes Separation Condition}}.{{harvnb|Pervin|1964|loc=p. 51}} 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 $[0,\; 1)$ and $(1,\; 2]$ are separated in the real line $\backslash Reals,$ 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 $d(p,\; q)\; \backslash geq\; r\; +\; s.$ The property of being separated can also be expressed in terms of derived set (indicated by the prime symbol): $A$ and $B$ are separated when they are disjoint and each is disjoint from the other's derived set, that is, $A\text{'}\; \backslash cap\; B\; =\; \backslash varnothing\; =\; B\text{'}\; \backslash cap\; A.$ (As in the case of the first version of the definition, the derived sets $A\text{'}$ and $B\text{'}$ are not required to be disjoint from each other.)

The sets $A$ and $B$ are {{visible anchor|separated by neighbourhoods}} if there are neighbourhoods $U$ of $A$ and $V$ 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. If $A$ and $B$ are open and disjoint, then they must be separated by neighbourhoods; just take $U\; =\; A$ and $V\; =\; B.$ For this reason, separatedness is often used with closed sets (as in the normal separation axiom).

The sets $A$ and $B$ are {{visible anchor|separated by closed neighbourhoods}} if there is 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 {{em|not}} separated by closed neighbourhoods. You could make either $U$ or $V$ 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 $A$ and $B$ are {{visible anchor|separated by a continuous function}} if there exists a continuous function $f\; :\; X\; \backslash to\; \backslash Reals$ from the space $X$ to the real line $\backslash Reals$ such that $A\; \backslash subseteq\; f^\{-1\}(0)$ and $B\; \backslash subseteq\; f^\{-1\}(1)$, that is, members of $A$ map to 0 and members of $B$ map to 1. (Sometimes the unit interval $[0,\; 1]$ is used in place of $\backslash Reals$ in this definition, but this makes no difference.) 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.{{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}} 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 $f$ as $U\; =\; f^\{-1\}[-c,\; c]$ and $V\; =\; f^\{-1\}[1\; -\; c,\; 1\; +\; c],$ where $c$ is any positive real number less than $1/2.$

The sets $A$ and $B$ are {{visible anchor|precisely separated by a continuous function}} if there exists a continuous function $f\; :\; X\; \backslash to\; \backslash Reals$ such that $A\; =\; f^\{-1\}(0)$ and $B\; =\; f^\{-1\}(1).$ (Again, you may also see the unit interval in place of $\backslash Reals,$ 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 $\backslash \{0\backslash \}$ and $\backslash \{1\backslash \}$ are closed in $\backslash Reals,$ 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).

{{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 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 usually called Hausdorff spaces or T₂ 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. 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 $\backslash emptyset$, authorities differ on whether $\backslash emptyset$ is connected and whether $\backslash emptyset$ is an open-connected component of itself.)

{{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 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.

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

{{reflist}}

{{refbegin}}

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

{{refend}}

{{Topology}}

Category:Separation axioms

Category:Topology