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Re: topology definition question
Posted:
Nov 17, 2012 3:47 AM
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Thank you very much Kaba and Jesse for your help. I appreciate it.
If it really does come down to convention, maybe for me it would be best that I just give all 4 of the criteria rather than have to first state what convention I am assuming for the set operations.
Thank you again,
Dan
On Saturday, November 17, 2012 10:33:04 AM UTC+8, Jesse F. Hughes wrote:
> Kaba writes:
>
>
>
> > 16.11.2012 23:42, Daniel J. Greenhoe wrote:
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> >> It seems the most "common" definition of a topology is that T is a topology on a set X if
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> >> 1. empty set is in T and
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> >> 2. X is in T and
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> >> 3. A and B are in T ==> A intersection B is in T and
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> >> 4. {A_i} in T ==> Union A_i is in T.
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> >>
>
> >> But some authors imply that only 3 and 4 are necessary for the definition of a topology. For example, Kelley ("General Topology", 1955, page 37) only uses 3 and 4 and says that these imply X is in T. McCarty ("Topology...", page 87) says 1 and 2 are "completely unneeded".
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> >>
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> >> My question is, is it really possible to exclude 1 and 2 from the definition such that 3 and 4 alone imply 1 and 2?
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> >>
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> >> Suppose X:={x,y,z} and T:={ {x},{y},{x,y} }.
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> >> Then T satisfies conditions 3 and 4, but yet X is not in T.
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> >> So how is it possible to exclude 3 from the definition of a topology?
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> >
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> > By convention, the intersection of zero number of subsets of X is the
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> > whole space X. Similarly, the union of zero number of subsets of X is
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> > the empty set.
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>
>
> Yes, but note that one needs to state (3) in terms of closure under all
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> finite intersections, rather than closure under binary intersections, in
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> order to ensure that (1) follows.
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>
>
> --
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> "I liked the world a lot better over ten years ago. I believed in a
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> lot more things. Hell, most people believed in a lot more things.
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> Back then the United States was still, well, known as most people used
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> to know the United States." -- James S. Harris in a nostalgic mood
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