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In the case of two dimensions, the intuition is as follows: For any line segment xy, consider the possible range of lengths of xv, where v is any point on the perpendicular bisector of xy. It is apparent that while there is no upper bound to the length of xv, there is a lower bound, which occurs when v is the midpoint of xy.
The arc length of a curve is defined as the least upper bound of the lengths of polygonal approximations. This generalization can be used to prove that the shortest curve between two points in Euclidean geometry is a straight line. No polygonal path between two points is shorter than the line between them.
More generally, one may define upper bound and least upper bound for any subset of a partially ordered set X, with “real number” replaced by “element of X ”. In this case, we say that X has the least-upper-bound property if every non-empty subset of X with an upper bound has a least upper bound in X.
The seldom-considered dual notion to a dcpo is the filtered-complete poset. Dcpos with a least element ("pointed dcpos") are one of the possible meanings of the phrase complete partial order (cpo). If every subset that has some upper bound has also a least upper bound, then the respective poset is called bounded complete. The term is used ...
The least-upper-bound property states that every nonempty subset of real numbers having an upper bound (or bounded above) must have a least upper bound (or supremum) in the set of real numbers. The rational number line Q does not have the least upper bound property. An example is the subset of rational numbers
A Saccheri quadrilateral is a quadrilateral which has two sides of equal length, both perpendicular to a side called the base. The other two angles of a Saccheri quadrilateral are called the summit angles and they have equal measure. The summit angles of a Saccheri quadrilateral are acute if the geometry is hyperbolic, and right angles if the ...
Then given any upper bound X of A 0, A 1, … in P(ω)/Fin, we can find a lesser upper bound, by removing from a representative for X one element of each a n. Therefore the A n have no supremum. Properties of complete Boolean algebras
Then has an upper bound (, for example, or ) but no least upper bound in : If we suppose is the least upper bound, a contradiction is immediately deduced because between any two reals and (including and ) there exists some rational , which itself would have to be the least upper bound (if >) or a member of greater than (if <).