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In the mathematical field of order theory, a continuum or linear continuum is a generalization of the real line. Formally, a linear continuum is a linearly ordered set ''S'' of more than one element that is densely ordered, i.e., between any two members there is another, and which "lacks gaps" in the sense that every non-empty subset with an upper bound has a least upper bound. More symbolically: a) ''S'' has the least-upper-bound property b) For each ''x'' in ''S'' and each ''y'' in ''S'' with ''x'' < ''y'', there exists ''z'' in ''S'' such that ''x'' < ''z'' < ''y'' A set has the least upper bound property, if every nonempty subset of the set that is bounded above has a least upper bound. Linear continua are particularly important in the field of topology where they can be used to verify whether an ordered set given the order topology is connected or not. Unlike the standard real line, a linear continuum may be bounded on either side: for example, any (real) closed interval is a linear continuum. ==Examples== * The ordered set of real numbers, R, with its usual order is a linear continuum, and is the archetypal example. Property b) is trivial, and property a) is simply a reformulaton of the completeness axiom. Examples in addition to the real numbers: *sets which are order-isomorphic to the set of real numbers, for example a real open interval, and the same with half-open gaps (note that these are not gaps in the above-mentioned sense) *the affinely extended real number system and order-isomorphic sets, for example the unit interval *the set of real numbers with only +∞ or only -∞ added, and order-isomorphic sets, for example a half-open interval *the long line * The set ''I'' × ''I'' (where × denotes the Cartesian product and ''I'' = (1 )) in the lexicographic order is a linear continuum. Property b) is trivial. To check property a), we define a map, π1 : ''I'' × ''I'' → ''I'' by: :''π''1 (''x'', ''y'') = ''x'' This map is known as the projection map. The projection map is continuous (with respect to the product topology on ''I'' × ''I'') and is surjective. Let ''A'' be a nonempty subset of ''I'' × ''I'' which is bounded above. Consider ''π''1(''A''). Since ''A'' is bounded above, ''π''1(''A'') must also be bounded above. Since, ''π''1(''A'') is a subset of ''I'', it must have a least upper bound (since ''I'' has the least upper bound property). Therefore, we may let ''b'' be the least upper bound of ''π''1(''A''). If ''b'' belongs to ''π''1(''A''), then ''b'' × ''I'' will intersect ''A'' at say ''b'' × ''c'' for some ''c'' ∈ ''I''. Notice that since ''b'' × ''I'' has the same order type of ''I'', the set (''b'' × ''I'') ∩ ''A'' will indeed have a least upper bound ''b'' × ''c, which is the desired least upper bound for ''A''. If ''b'' doesn't belong to ''π''1(''A''), then ''b'' × 0 is the least upper bound of ''A'', for if ''d'' < ''b'', and ''d'' × ''e'' is an upper bound of ''A'', then ''d'' would be a smaller upper bound of ''π''1(''A'') than ''b'', contradicting the unique property of ''b''. 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「Linear continuum」の詳細全文を読む スポンサード リンク
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