ONT Re: Topology
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| 1. Topological Spaces
|
| 1.3. Accumulation Points
|
| The topology of a topological space can be described in terms of
| neighborhoods of points and consequently it must be possible to
| formulate a description of closed sets in terms of neighborhoods.
| This formulation leads to a new classification of points in the
| following way. A set A is closed iff X~A is open, and hence iff
| each point of X~A has a neighborhood which is contained in X~A,
| or, equivalently, is disjoint from A. Consequently, A is closed
| iff for each x, if every neighborhood of x intersects A, then x
| is in A. This suggests the following definition.
|
| A point x is an 'accumulation point' (sometimes called
| 'cluster' point or 'limit' point) of a subset A of a
| topological space (X,!T!) iff every neighborhood
| of x contains points of A other than x. Then it
| is true that each neighborhood of a point x
| intersects A if and only if x is either a
| point of A or an accumulation point of A.
| The following theorem is then clear.
|
| 5. Theorem. A subset of a topological space
| is closed if and only if it contains the
| set of its accumulation points.
|
| JLK, Gen Top, pages 40-41.
|
| John L. Kelley, 'General Topology',
| Van Nostrand Reinhold, New York, NY, 1955.
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