Editing Empty set (section)

==In other areas of mathematics==
=== 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.

=== Topology ===
In any [[topological space]] <math>X</math>, the empty set is [[open set|open]] by definition, as is <math>X</math>. Since the [[Complement (set theory)|complement]] of an open set is [[Closed set|closed]] and the empty set and <math>X</math> are complements of each other, the empty set is also closed, making it a [[clopen set]]. Moreover, the empty set is [[Compact set|compact]] by the fact that every [[finite set]] is compact.

A topological space <math>X</math> is said to have the [[indiscrete topology]] if the only open sets are <math>\varnothing</math> and the entire space.

The [[Closure (mathematics)|closure]] of the empty set is empty. This is known as "preservation of [[nullary]] [[Union (set theory)|unions]]".<ref>{{cite book |last1=Munkres |first1=James Raymond |title=Topology |date=2018 |publisher=Pearson |location=New York, NY |isbn=978-0134689517 |edition=Second, reissue}}</ref>

=== Category theory ===
If <math>A</math> is a set, then there exists precisely one [[Function (mathematics)|function]] <math>f</math> from <math>\varnothing</math> to <math>A,</math> the [[empty function]]. As a result, the empty set is the unique [[initial object]] of the [[Category theory|category]] of sets and functions.

The empty set can be turned into a [[topological space]], called the empty space, in just one way: by defining the empty set to be [[Open set|open]]. This empty topological space is the unique initial object in the [[category of topological spaces]] with [[Continuous function (topology)|continuous maps]]. In fact, it is a [[strict initial object]]: only the empty set has a function to the empty set.

=== Set theory ===
In the [[von Neumann ordinal|von Neumann construction of the ordinals]], 0 is defined as the empty set, and the successor of an ordinal is defined as <math>S(\alpha)=\alpha\cup\{\alpha\}</math>. Thus, we have <math>0=\varnothing</math>, <math>1 = 0\cup\{0\}=\{\varnothing\}</math>, <math>2=1\cup\{1\}=\{\varnothing,\{\varnothing\}\}</math>, and so on.  The von Neumann construction, along with the [[axiom of infinity]], which guarantees the existence of at least one infinite set, can be used to construct the set of natural numbers, <math>\N_0</math>, such that the [[Peano axioms]] of arithmetic are satisfied.