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, order and/or properties:
“In England if something goes wrong—say, if one finds a skunk in the garden—he writes to the family solicitor, who proceeds to take the proper measures; whereas in America, you telephone the fire department. Each satisfies a characteristic need; in the English, love of order and legalistic procedure; and here in America, what you like is something vivid, and red, and swift.”
—Alfred North Whitehead (1861–1947)
“We would not let ourselves be burned at the stake for our opinions: we are not that certain of them. But perhaps we would do so in order to be allowed to possess and to change them.”
—Friedrich Nietzsche (1844–1900)
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)