ONT Re: Topology
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| 1. Topological Spaces
|
| 1.6. Bases and Subbases (cont.)
|
| We have just seen that an arbitrary family !S! of sets may fail to be the base
| for any topology. With admirable persistence we vary the question and enquire
| whether there is a unique topology which is, in some sense, generated by !S!.
| Such a topology should be a topology for the set X which is the union of the
| members of !S!, and each member of !S! should be open relative to the topology;
| that is, !S! should be a subfamily of the topology. This raises the question:
| Is there a smallest topology for X which contains !S!? The following simple
| result will enable us to exhibit this smallest topology.
|
| 12. Theorem. If !S! is any non-void family of sets
| the family of all finite intersections of members
| of !S! is the base for a topology for the set
| X = |_|{S : S in !S!}.
|
| Proof. If !S! is a family of sets let !B! be the family of
| finite intersections of members of !S!. Then the intersection
| of two members of !B! is again a member of !B! and, applying the
| preceding theorem, !B! is the base for a topology. þ
|
| A family !S! of sets is a 'subbase for a topology' !T! iff
| the family of finite intersections of members of !S! is a
| base for !T! (equivalently, iff each member of !T! is the
| union of finite intersections of members of !S!). In view
| of the preceding theorem every non-empty family !S! is the
| subbase for some topology, and this topology is, of course,
| uniquely determined by !S!. It is the smallest topology
| containing !S! (that is, it is a topology containing !S!
| and is a subfamily of every topology containing !S!).
|
| There will generally be many different bases and subbases
| for a topology and the most appropriate choice may depend on
| the problem under consideration. One rather natural subbase
| for the usual topology for the real numbers is the family of
| half-infinite open intervals; that is, the family of sets of
| the form {x : x > a} or {x : x < a}. Each open interval is the
| intersection of two such sets, and this family is consequently a
| subbase. The family of all sets of the same form with 'a' rational
| is a less obvious and more interesting subbase. (See problem 1.J.)
|
| JLK, Gen Top, pages 47-48.
|
| John L. Kelley, 'General Topology',
| Van Nostrand Reinhold, New York, NY, 1955.
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