Extended natural numbers

In mathematics, the extended natural numbers is a set which contains the values $$0, 1, 2, \dots$$ and $$\infty$$ (infinity). That is, it is the result of adding a maximum element $$\infty$$ to the natural numbers. Addition and multiplication work as normal for finite values, and are extended by the rules $$n+\infty=\infty+n=\infty$$ ($$n\in\mathbb{N}\cup \{\infty\}$$), $$0\times \infty=\infty \times 0=0$$ and $$m\times \infty=\infty\times m=\infty$$ for $$m\neq 0$$.

With addition and multiplication, $$\mathbb{N}\cup \{\infty\}$$ is a semiring but not a ring, as $$\infty$$ lacks an additive inverse. The set can be denoted by $$\overline{\mathbb{N}}$$, $$\mathbb{N}_\infty$$ or $$\mathbb{N}^\infty$$. It is a subset of the extended real number line, which extends the real numbers by adding $$-\infty$$ and $$+\infty$$.

Applications
In graph theory, the extended natural numbers are used to define distances in graphs, with $$\infty$$ being the distance between two unconnected vertices. They can be used to show the extension of some results, such as the max-flow min-cut theorem, to infinite graphs.

In topology, the topos of right actions on the extended natural numbers is a category PRO of projection algebras.

In constructive mathematics, the extended natural numbers $$\mathbb{N}_\infty$$ are a one-point compactification of the natural numbers, yielding the set of non-increasing binary sequences i.e. $$(x_0,x_1,\dots)\in 2^\mathbb{N}$$ such that $$\forall i\in\mathbb{N}: x_i\ge x_{i+1}$$. The sequence $$1^n 0^\omega$$ represents $$n$$, while the sequence $$1^\omega$$ represents $$\infty$$. It is a retract of $$2^\mathbb{N}$$ and the claim that $$\mathbb{N}\cup \{\infty\}\subseteq \mathbb{N}_\infty$$ implies the limited principle of omniscience.