ONT Re: Topology
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| 1. Topological Spaces
|
| 1.1. Topologies and Neighborhoods (cont.)
|
| The set of real numbers, with an appropriate topology, is
| a very interesting topological space. This is scarcely
| surprising since the notion of a topological space is
| an abstraction of some interesting properties of the
| real numbers. The 'usual topology' for the real
| numbers is the family of all those sets which
| contain an open interval about each of their
| points. That is, a subset A of the set of
| real numbers is open iff for each member x
| of A there are numbers a and b such that
| a < x < b and the 'open interval'
| {y : a < y < b} is a subset of A.
| Of course, we must verify that
| this family of sets is indeed
| a topology, but this offers
| no difficulty. It is worth
| noticing that, conveniently,
| an open interval is an
| open set.
|
| JLK, Gen Top, page 38.
|
| John L. Kelley, 'General Topology',
| Van Nostrand Reinhold, New York, NY, 1955.
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