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Topic: topology definition question
Replies: 1   Last Post: Nov 16, 2012 4:56 PM

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Kaba

Posts: 289
Registered: 5/23/11
Re: topology definition question
Posted: Nov 16, 2012 4:56 PM
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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.

--
http://kaba.hilvi.org



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