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

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Jesse F. Hughes

Posts: 9,776
Registered: 12/6/04
Re: topology definition question
Posted: Nov 16, 2012 9:31 PM
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Kaba <kaba@nowhere.com> 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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