quasi
Posts:
12,021
Registered:
7/15/05


Re: compact
Posted:
May 15, 2013 4:48 PM


dullrich wrote: >William Elliot wrote: >> A point x is an accumulation point of A >> when for all open U nhood x, U /\ A is infinite. >> >> If S is compact, then every infinite set A has an >> accumulation point. >> >> Proof. >> If not, then for all x, there's some open U_x nhood x with >> finite U /\ A. Since C = { U_x  x in S } covers S, there's >> a finite subcover { U_x1,.. U_xj } >> Thus A = A /\ (U_x1 \/..\/ U_xj } >> = (U_x1 /\ A) \/..\/ (U_xj /\ A), >> is finite, > >"has cardinality less than A", not "is finite".
Looks like a finite union of finite sets, no?
>> which of course it isn't.
quasi

