ONT Re: Topology
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| 1. Topological Spaces
|
| 1.6. Bases and Subbases (cont.)
|
| A set A is 'dense' in a topological space X iff the closure of A is X.
| A topological space X is 'separable' iff there is a countable subset which
| is dense in X. A separable space may fail to satisfy the second axiom of
| countability. For example, let X be an uncountable set with the topology
| consisting of the void set and the complements of finite sets. Then every
| non-finite set is dense because it intersects every non-void open set. On
| the other hand, suppose that there is a countable base !B! and let x be a
| fixed point of X. The intersection of the family of all open sets to which
| x belongs must be {x}, because the complement of every other point is open.
| It follows that the intersection of those members of the base !B! to which
| x belongs is {x}. But the complement of this countable intersection is the
| union of a countable number of finite sets, hence countable, and this is a
| contradiction. (Less trivial examples occur later.) There is no difficulty
| in showing that a space with a countable base is separable.
|
| 14. Theorem. A space whose topology has a countable base is separable.
|
| Proof. Choose a point out of each member of the base, thus obtaining
| a countable set A. The complement of the closure of A is an open set
| which, being disjoint from A, contains no non-void member of the base
| and is hence void. þ
|
| JLK, Gen Top, page 49.
|
| John L. Kelley, 'General Topology',
| Van Nostrand Reinhold, New York, NY, 1955.
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