ONT Re: Topology
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| 1. Topological Spaces
|
| 1.3. Accumulation Points (cont.)
|
| If x is an accumulation point of A it is sometimes said, in a pleasantly
| suggestive phrase, that there are points of A arbitrarily near x. If we
| pursue this imagery it appears that an indiscrete topological space is
| really quite crowded, for each point x is an accumulation point of every
| set other than the void set and the set {x}. On the other hand, in a
| discrete topological space, no point is an accumulation point of a set.
|
| If X is the set of real numbers with the usual topology a
| variety of situations can arise. If A is the open interval
| (0, 1), then every point of the closed interval [0, 1] is an
| accumulation point of A. If A is the set of all non-negative
| rationals with squares less than 2, then the closed interval
| [0, 2^½] is the set of accumulation points. If A is the set
| of all reciprocals of integers, then 0 is the only accumulation
| point of A, and the set of integers has no accumulation points.
|
| 6. Theorem. The union of a set and the
| set of its accumulation points is closed.
|
| Proof. If x is neither a point nor accumulation point of A, then there
| is an open neighborhood U of x which does not intersect A. Since U is
| a neighborhood of each of its points, no one of these is an accumulation
| point of A. Hence the union of the set A and the set of its accumulation
| points is the complement of an open set. þ
|
| The set of all accumulation points of a set A
| is sometimes called the 'derived' set of A.
|
| JLK, Gen Top, pages 41-42.
|
| John L. Kelley, 'General Topology',
| Van Nostrand Reinhold, New York, NY, 1955.
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