• Sep 14th 2011, 02:46 PM
paupsers
Theorem. Let $D \subset Y$. Suppose that D is non-empty and sequentially compact. Let C be an open cover of D. Then there is a real number $\epsilon > 0$ such that if $E \subset D$ with $d(E)<\epsilon$ then $E \subset A_{\alpha}$ for some $A_{\alpha}\in C$.
Note that d(E) is the diameter of E.

This seems so intuitively obvious to me... or am I not reading it correctly? Can't I just choose $\epsilon$ small enough that E is approximately a point?
• Sep 14th 2011, 03:36 PM
Plato
Theorem. Let $D \subset Y$. Suppose that D is non-empty and sequentially compact. Let C be an open cover of D. Then there is a real number $\epsilon > 0$ such that if $E \subset D$ with $d(E)<\epsilon$ then $E \subset A_{\alpha}$ for some $A_{\alpha}\in C$. Note that d(E) is the diameter of E.
This seems so intuitively obvious to me... or am I not reading it correctly? Can't I just choose $\epsilon$ small enough that E is approximately a point?
Suppose not. If $n\in\mathbb{Z}^+$ there must be a set $B_n$ such that $0 but $B_n$ is a subset of no open set in the cover.