Could someone give an example of
a) a non-compact space with an infinite subset which has a limit point
With the euclidean topology inherited from $\displaystyle \mathbb{R}$ , the set $\displaystyle [0,1)$ and the set $\displaystyle \left\{\frac{1}{n}\right\}$
b) A non-compact space with an infinite subset which has a no limit point
Do a very, VERY slight modification to the above example... Tonio