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
Famous quotes containing the words order and/or properties:
“In order to become spoiled ... a child has to be able to want things as well as need them. He has to be able to see himself as a being who is separate from everyone else.... A baby is none of these things. He feels a need and he expresses it. He is not intellectually capable of working out involved plans and ideas like “Can I make her give me...?” “If I make enough fuss he will...?” “They let me do ... yesterday and I want to do it again today so I’ll....””
—Penelope Leach (20th century)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)