Order and Topological Properties
The affinely extended real number system turns into a totally ordered set by defining −∞ ≤ a ≤ +∞ for all a. This order has the desirable property that every subset has a supremum and an infimum: it is a complete lattice.
This induces the order topology on R. In this topology, a set U is a neighborhood of +∞ if and only if it contains a set {x : x > a} for some real number a, and analogously for the neighborhoods of −∞. R is a compact Hausdorff space homeomorphic to the unit interval . Thus the topology is metrizable, corresponding (for a given homeomorphism) to the ordinary metric on this interval. There is no metric which is an extension of the ordinary metric on R.
With this topology the specially defined limits for x tending to +∞ and −∞, and the specially defined concepts of limits equal to +∞ and −∞, reduce to the general topological definitions of limits.
Read more about this topic: Extended Real Number Line
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