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Re: topology definition question
Posted:
Nov 17, 2012 11:14 AM


"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



