In physics, approximations of real numbers are used for extended real number system, or by requiring the counting of an infinite number of events.
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The affinely extended real number system turns into a totally ordered set by defining-\ infty \ leq a \ leq + \ infty for all a.
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Examples are the closed intervals of real numbers, e . g . the unit interval [ 0, 1 ], and the affinely extended real number system ( extended real number line ).
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Since the supremum and infimum of an unbounded set of real numbers may not exist ( the reals are not a complete lattice ), it is convenient to consider sequences in the affinely extended real number system : we add the positive and negative infinities to the real line to give the complete totally ordered set ( " ", " ), which is a complete lattice.
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The limit of a sequence of points \ left ( x _ n : n \ in \ mathbb { N } \ right ) \; in a topological space " T " is a special case of the limit of a function : the domain is \ mathbb { N } in the space \ mathbb { N } \ cup \ lbrace + \ infty \ rbrace with the induced topology of the affinely extended real number system, the range is " T ", and the function argument " n " tends to + ", which in this space is a limit point of \ mathbb { N }.
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