A space is said to be Dedekind   complete if every bounded subset of it contains one of these, and the set of rationals whose squares are less than two does not have one of these in the rationals.
The essential versions of these hold their desired properties everywhere but a set of measure zero. A lattice is a partially ordered set all of whose  nite subsets            have these, and if the lattice is complete, then all of its subsets have these. If a linearly ordered set has the property named for this concept, and it is dense, it is called a linear continuum. A space is said to be Dedekind   complete if every bounded subset of it contains one of these, and the set of rationals whose squares are less than two does not have one of these in the rationals. For ten points, identify this analytic concept which is the least upper bound of a set.
The essential versions of these hold their desired properties everywhere but a set of measure zero.
supremum [accept in mum until  least upper bound  because the clues are logically equivalent, accept least upper bound before read, accept greatest lower bound until  least upper bound  is read, do not accept or prompt on  maximum  or  minimum ]