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In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. Some of these restrictions are given by the separation axioms. These are sometimes called ''Tychonoff separation axioms'', after Andrey Tychonoff. The separation axioms are axioms only in the sense that, when defining the notion of topological space, one could add these conditions as extra axioms to get a more restricted notion of what a topological space is. The modern approach is to fix once and for all the axiomatization of topological space and then speak of ''kinds'' of topological spaces. However, the term "separation axiom" has stuck. The separation axioms are denoted with the letter "T" after the German ''Trennungsaxiom'', which means "separation axiom." The precise meanings of the terms associated with the separation axioms has varied over time, as explained in History of the separation axioms. It is important to understand the authors' definition of each condition mentioned to know exactly what they mean, especially when reading older literature. ==Preliminary definitions== Before we define the separation axioms themselves, we give concrete meaning to the concept of separated sets (and points) in topological spaces. (Separated sets are not the same as ''separated spaces'', defined in the next section.) The separation axioms are about the use of topological means to distinguish disjoint sets and distinct points. It's not enough for elements of a topological space to be distinct (that is, unequal); we may want them to be ''topologically distinguishable''. Similarly, it's not enough for subsets of a topological space to be disjoint; we may want them to be ''separated'' (in any of various ways). The separation axioms all say, in one way or another, that points or sets that are distinguishable or separated in some weak sense must also be distinguishable or separated in some stronger sense. Let ''X'' be a topological space. Then two points ''x'' and ''y'' in ''X'' are ''topologically distinguishable'' if they do not have exactly the same neighbourhoods (or equivalently the same open neighbourhoods); that is, at least one of them has a neighbourhood that is not a neighbourhood of the other (or equivalently there is an open set that one point belongs to but the other point does not). Two points ''x'' and ''y'' are ''separated'' if each of them has a neighbourhood that is not a neighbourhood of the other; that is, neither belongs to the other's closure. More generally, two subsets ''A'' and ''B'' of ''X'' are ''separated'' if each is disjoint from the other's closure. (The closures themselves do not have to be disjoint.) All of the remaining conditions for separation of sets may also be applied to points (or to a point and a set) by using singleton sets. Points ''x'' and ''y'' will be considered separated, by neighbourhoods, by closed neighbourhoods, by a continuous function, precisely by a function, if and only if their singleton sets and are separated according to the corresponding criterion. Subsets ''A'' and ''B'' are ''separated by neighbourhoods'' if they have disjoint neighbourhoods. They are ''separated by closed neighbourhoods'' if they have disjoint closed neighbourhoods. They are ''separated by a continuous function'' if there exists a continuous function ''f'' from the space ''X'' to the real line R such that the image ''f''(''A'') equals and ''f''(''B'') equals . Finally, they are ''precisely separated by a continuous function'' if there exists a continuous function ''f'' from ''X'' to R such that the preimage ''f''−1() equals ''A'' and ''f''−1() equals ''B''. These conditions are given in order of increasing strength: Any two topologically distinguishable points must be distinct, and any two separated points must be topologically distinguishable. Any two separated sets must be disjoint, any two sets separated by neighbourhoods must be separated, and so on. For more on these conditions (including their use outside the separation axioms), see the articles Separated sets and Topological distinguishability. 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「Separation axiom」の詳細全文を読む スポンサード リンク
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