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In topology and related branches of mathematics, separated sets are pairs of subsets 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 spaces (and their connected components) as well as to the separation axioms for topological spaces. Separated sets should not be confused with separated spaces (defined below), which are somewhat related but different. Separable spaces are again a completely different topological concept. == Definitions == There are various ways in which two subsets of a topological space ''X'' can be considered to be separated. *''A'' and ''B'' 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 do not have to be disjoint from each other; for example, the intervals (and (1,2 ) are separated in the real line R, even though the point 1 belongs to both of their closures. More generally in any metric space, two open balls ''B''''r''(x1) = and ''B''''s''(x2) = are separated whenever ''d''(x1,x2) ≥ ''r''+''s''. Note that any two separated sets automatically must be disjoint. *''A'' and ''B'' are 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'' = (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). *''A'' and ''B'' are 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, (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 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. *''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'') = and ''f''(''B'') = . (Sometimes you will see the unit interval () used in place of R in this definition, but it makes no difference in the end.) In our example, (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() and ''V'' := ''f''−1(), 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 and 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). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Separated sets」の詳細全文を読む スポンサード リンク
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