ONT Re: Topology
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| 1. Topological Spaces
|
| 1.6. Bases and Subbases (cont.)
|
| A family !A! is a 'cover' of a set B iff B is a subset of the
| union |_|{A : A in !A!}, that is, iff each member of B belongs
| to some member of !A!. The family is an 'open cover' of B iff
| each member of !A! is an open set. A 'subcover' of !A! is a
| subfamily which is also a cover.
|
| 15. Theorem (Lindelöf). There is a countable subcover of each open
| cover of a subset of a space whose topology has a countable base.
|
| Proof. Suppose A is a set, !A! is an open cover of A, and !B!
| is a countable base for the topology. Because each member of !A!
| is the union of members of !B! there is a subfamily !C! of !B! which
| also covers A, such that each member of !C! is a subset of some member
| of !A!. For each member of !C! we may select a containing member of !A!
| and so obtain a countable subfamily !D! of !A!. Then !D! is also a cover
| of A because !C! covers A. Hence !A! has countable subcover. þ
|
| A topological space is a 'Lindelöf space' iff each
| open cover of the space has a countable subcover.
|
| JLK, Gen Top, pages 49-50.
|
| John L. Kelley, 'General Topology',
| Van Nostrand Reinhold, New York, NY, 1955.
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