Date: May 15, 2013 12:15 AM
Author: William Elliot
Subject: compact
A point x is an axxumulation 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, which of course it isn't.

If every infinite A subset S has an accumulation point, is S compact?