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Topic: topology definition question
Replies: 5   Last Post: Nov 17, 2012 11:14 AM

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Achimota

Posts: 254
Registered: 4/30/07
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:
>
> >> It seems the most "common" definition of a topology is that T is a topology on a set X if
>
> >> 1. empty set is in T and
>
> >> 2. X is in T and
>
> >> 3. A and B are in T ==> A intersection B is in T and
>
> >> 4. {A_i} in T ==> Union A_i is in T.
>
> >>
>
> >> 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".
>
> >>
>
> >> 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?
>
> >>
>
> >> Suppose X:={x,y,z} and T:={ {x},{y},{x,y} }.
>
> >> Then T satisfies conditions 3 and 4, but yet X is not in T.
>
> >> So how is it possible to exclude 3 from the definition of a topology?
>
> >
>
> > By convention, the intersection of zero number of subsets of X is the
>
> > whole space X. Similarly, the union of zero number of subsets of X is
>
> > the empty set.
>
>
>
> Yes, but note that one needs to state (3) in terms of closure under all
>
> finite intersections, rather than closure under binary intersections, in
>
> order to ensure that (1) follows.
>
>
>
> --
>
> "I liked the world a lot better over ten years ago. I believed in a
>
> lot more things. Hell, most people believed in a lot more things.
>
> Back then the United States was still, well, known as most people used
>
> to know the United States." -- James S. Harris in a nostalgic mood






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