
Re: Another AC anomaly?
Posted:
Dec 17, 2009 10:01 AM


In article <876387t8ds.fsf@phiwumbda.org> "Jesse F. Hughes" <jesse@phiwumbda.org> writes: ... > In general, the term limit is defined topologically: any topological > space comes with its own natural definition of limit. But it's not at > all clear to me whether "general set convergence" is a limit in this > particular sense. Is there a topology on Set so that the "natural" > definition of limit coincides with general set convergence?
I do not think there is a proper topology, but I do not know.
> If not, then I guess K_h has a point about the naturalness of this > definition of limit (and, conversely, if so, then K_h has no good > point at all).
But here I think different. Within the definition presented there is a limit definition for sets that is derived from the underlying topology on the elements of those sets. At such it appears to me to be a natural definition, that is, an element is also element of the lim sup, it should be a limit point of one of the many pointwise sequences you can create when you take from each of the sets an element. lim inf consists of those elements that are limits of such pointwise sequences.
And with the discrete topology on the elements we get just the definitions I quoted. So when we assume the discrete topology on N, we get lim sup(n > oo) {n} = {} because there is no limit point. With the set "N union {oo}" with a topology based on delta(n, m) = 1/m  1/n we get a different result: lim sup(n > oo) {n} = {oo}.
K_h would like that if each S_n of a sequence is a set with a single element, that being a set exain (e.g. {X_n}) is that lim [supinf] S_n = {lim [supinf] X_n} but for that we need a more complicated (and in my opinion less natural) version of the limit of a sequence of sets, just because there is no proper topology on the collection of X_n.  dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/

