Editing Empty set (section)

=== Extended real numbers ===

Since the empty set has no member when it is considered as a subset of any [[ordered set]], every member of that set will be an upper bound and lower bound for the empty set. For example, when considered as a subset of the real numbers, with its usual ordering, represented by the [[real number line]], every real number is both an upper and lower bound for the empty set.<ref>Bruckner, A.N., Bruckner, J.B., and Thomson, B.S. (2008). ''[http://classicalrealanalysis.info/com/documents/TBB-AllChapters-Portrait.pdf Elementary Real Analysis]'', 2nd edition, p.&nbsp;9.</ref> When considered as a subset of the [[extended reals]] formed by adding two "numbers" or "points" to the real numbers (namely [[negative infinity]], denoted <math>-\infty\!\,,</math> which is defined to be less than every other extended real number, and [[positive infinity]], denoted <math>+\infty\!\,,</math> which is defined to be greater than every other extended real number), we have that: 
<math display=block>\sup\varnothing=\min(\{-\infty, +\infty \} \cup \mathbb{R})=-\infty,</math>
and
<math display=block>\inf\varnothing=\max(\{-\infty, +\infty \} \cup \mathbb{R})=+\infty.</math>

That is, the least upper bound (sup or [[supremum]]) of the empty set is negative infinity, while the greatest lower bound (inf or [[infimum]]) is positive infinity. By analogy with the above, in the domain of the extended reals, negative infinity is the identity element for the maximum and supremum operators, while positive infinity is the identity element for the minimum and infimum operators.