If X is a compact metric space and A a part of X. The definition of locally finite for A is: For all $x\in$X, $\exists \epsilon$ such as $B(x,\epsilon)$ only contains a finite number of points of A.
My question is: does this still stands if by "For all x$\in$X" we write "For all x$\in$A"?
I think it does not but i can't seem to prove it