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

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Frederick Williams

Posts: 2,164
Registered: 10/4/10
Re: topology definition question
Posted: Nov 17, 2012 11:14 AM
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"Daniel J. Greenhoe" wrote:
>
> 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.


That the union of the empty set of subsets of S is the empty set, and
that the intersection of the empty set of subsets of S is S, is not just
convention. Consider:

let I be an index set, and let {A_i: i in I} be a set of subsets of S,
then

bigcup_{i in I} A_i =df {x in S: x in A_j for some j in I} (1)

and

bigcap_{i in I} A_i =df {x in S: x in A_j for all j in I}. (2)

So _by those definitions_

bigcup_{i in emptyset} A_i = emptyset, (3)

and

bigcap_{i in emptyset} A_i = S. (4)

To be more precise, if (3) and (4) are just so by convention, then the
conventions in question are (1) and (2); and (1) and (2) are obvious
generalizations of the definitions of small cup and cap. (1) and (2)
have to be generalizations of the definitions of small cup and cap in
order that when I is a two element set, bigcup means cup and bigcap
means cap.

In another post you refer to Mendelson (I assume 'Introduction to
topology'), see section 4 of chapter 1 in that work.

--
When a true genius appears in the world, you may know him by
this sign, that the dunces are all in confederacy against him.
Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting



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