ONT Re: Topology
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| 1. Topological Spaces
|
| 1.1. Topologies and Neighborhoods (cont.)
|
| A set U in a topological space (X, !T!) is a 'neighborhood' (!T!-neighborhood)
| of a point x iff U contains an open set to which x belongs. A neighborhood of
| a point need not be an open set, but every open set is a neighborhood of each
| of its points. Each neighborhood of a point contains an open neighborhood
| of the point. If !T! is the indiscrete topology the only neighborhood of
| a point x is the space X itself. If !T! is the discrete topology, then
| every set to which a point belongs is a neighborhood of it. If X is the
| set of real numbers and !T! is the usual topology, then a neighborhood of
| a point is a set containing an open interval to which the point belongs.
|
| 1. Theorem. A set is open if and only if
| it contains a neighborhood of each of
| its points.
|
| Proof. The union U of all open subsets of a set A is surely an open subset of A.
| If A contains a neighborhood of each of its points, then each member x of A belongs
| to some open subset of A and hence x is in U. In this case A = U and therefore A
| is open. On the other hand, if A is open it contains a neighborhood (namely, A)
| of each of its points. þ
|
| The foregoing theorem evidently implies that a set is
| open iff it is a neighborhood of each of its points.
|
| The 'neighborhood system' of a point is the family
| of all neighborhoods of the point.
|
| 2. Theorem. If !U! is the neighborhood system of a point,
| then finite intersections of members of !U! belong to !U!,
| and each set which contains a member of !U! belongs to !U!.
|
| Proof. If U and V are neighborhoods of a point x, there are
| open neighborhoods U_0 and V_0 contained in U and V respectively.
| Then U |^| V contains the open neighborhood U_0 |^| V_0 and is hence
| a neighborhood of x. Thus the intersection of two (and hence of any
| any finite number of) members of !U! is a member. If a set U contains
| a neighborhood of a point x it contains an open neighborhood of x and is
| consequently itself a neighborhood. þ
|
| JLK, Gen Top, pages 38-39.
|
| John L. Kelley, 'General Topology',
| Van Nostrand Reinhold, New York, NY, 1955.
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