Extended natural numbers
In graph theory, the extended natural numbers are used to define distances in graphs, with formula_2 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 formula_12 are a one-point compactification of the natural numbers, yielding the set of non-increasing binary sequences i.e. formula_18 such that formula_19. The sequence formula_20 represents formula_21, while the sequence formula_22 represents formula_2. It is a retract of formula_24 and the claim that formula_25 implies the limited principle of omniscience.
Applications
In mathematics, the extended natural numbers is a set which contains the values formula_1 and formula_2 (infinity). That is, it is result of adding a maximum element formula_2 to the natural numbers. The set can be denoted by formula_11, formula_12 or formula_13.