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Long lineIn topology, the long line is a topological space analogous to the real line, only much longer. Because it behaves locally just like the real line, but has different large-scale properties, it serves as one of the basic counterexamples of topology.
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We start with the first uncountable ordinal ω1. This is a totally ordered set, and the cartesian product ω1 × [0, 1) becomes a totally ordered set if we use the lexicographic or dictionary order[?]. The long line L is defined as ω1 × [0, 1) with the order topology arising from this total order. That is, it consists of an uncountable number of copies of [0, 1) 'pasted together' end-to-end. Compare this with the real interval [0, ∞), which can be viewed as a countable number of copies of [0, 1) pasted together end-to-end. A related space, the extended long line, L*, is obtained by adjoining an additional element to the end of L.
Both L and L* are normal Hausdorff spaces because they are order topologies. Both of them have the same cardinality as the real line, yet they are 'much longer'. Both of them are locally compact. Neither of them is metrisable.
The long line L isn't paracompact. It is path-connected and simply connected but not contractible. L is a one-dimensional topological manifold with boundary. L is first countable but not second countable.
The extended long line L* is compact; it is the one-point compactification of L. It is also connected, but not path-connected because the long line is 'too long' to be covered by a path, which is an image of an interval. L* isn't a manifold and isn't first countable.
In telephone systems nomenclature a long line is a transmission line in a long-distance communications network such as carrier systems, microwave radio links, geosynchronous satellite links, underground cables, aerial cables and open wire, and submarine cables. In the United States, some of this technology was spun-off into the corporate entity known as AT&T Long Distance with the breakup of AT&T.
"The present period appears to have been one of the happiest in the life
brothers, and their Princesses, with the King's saint-like sister
dined in public. These ties seemed to be drawn daily closer for some
Comtesse d'Artois lay-in, the Dauphine, then become Queen, transferred
gratification of her society.
"During all this time, however, Du Barry, the Duc d'Aiguillon, and the
any tenderness on the part of the husband Dauphin, and, from different
hatred sent back to Vienna."
anxiety, not only about her daughter, but her State policy, which it may
the machinations of an individual who must now appear upon the stage of
Marie Antoinette.
"I allude to the Cardinal Prince de Rohan.
"At this time he was Ambassador at the Court of Vienna. The reliance the
daughter's reputation. He was the cause of her sending spies to watch
cultivate, as well as of those it was deemed desirable for her to exclude
secretly received pensions from Vienna, she could, of course, tell,
Dauphine was warned that she was surrounded by persons who were not her
will sufficiently explain how much the Empress must have been chagrined
which was intended to have been served in sending her to France. A less.html">less
mother. Marie Antoinette had much less of the politician about her than
amusement, that she never even thought of entering into State affairs
and called upon by the Ministers themselves to screen them from
seat in the Cabinet Council (though she took it with great reluctance)
its necessity. She would often exclaim to me: 'How happy I was during
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