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Topic:
compact
Replies:
9
Last Post:
May 18, 2013 9:56 PM



quasi
Posts:
12,067
Registered:
7/15/05


Re: compact
Posted:
May 16, 2013 3:02 PM


dullrich wrote: >quasi wrote: >>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? > >A typo, sort of. Later in the same post he said >"is finite", where he should have said "has >cardinality less than A". So I decided to post >a comment, and then I inserted the comment in >the wrong place, this paragraph looking so much >like that paragraph...
Had I read the rest of the post (after your comment), I might have realized (as William Elliot did) that you had simply accidentally misplaced your comment.
quasi



