September 4th, 2010, 11:34 AM  #1  
Member Joined: Nov 2009 Posts: 72 Thanks: 0  Topology generated from subbasis
I'm teaching myself topology from Munkres' book, and I ran across this question in the exercises: Quote:
 
September 4th, 2010, 12:48 PM  #2 
Senior Member Joined: Jun 2010 Posts: 618 Thanks: 0  Re: Topology generated from subbasis
Hello martexel. I think the question is asking you to prove that if A is a basis for a topology on X, then the topology generated by A equals the intersection of all the topologies on X containing A, and if alternatively A is a subbasis for a topology on X, then the topology generated by A equals the intersection of all the topologies on X containing A. Therefore, it is not implicit in the question that the two topologies so generated be equal, since A as a basis in the first case and as a subbasis in the second are in no way related. It may help to subscript the two differently. The crux of the matter is how we define "the topology generated by a basis" versus "the topology generated by a subbasis", as well as the difference in the definition of "basis" and "subbasis". (Keep in mind that a basis is automatically a subbasis, so a subbasis is "easier" to produce.) Look over these differences again and your worries will disappear. Regards, Ormkärr 

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generated, subbasis, topology 
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