what is the difference between isolated point and non isolated point in metric space X? Im thinking that  example  of isolated point  are  $x \in\mathbb{N}$ and $x \in\mathbb{Z}$ because  isolated point are discrete
example of non -isolated point are $ x\in \mathbb{R}, x \in \mathbb{Q}$ , $x \in \mathbb{R}-\mathbb{Q}$
Is its  true ?
 A: An isolated point in a metric space is a point with some open ball around it, in which there is no other member of the space. This depends on not only the point, but the space itself.
It's not clear whether or not your example is correct, since you did not specify the metric space. If your metric space is the real line, then both examples describe non-isolated points, since you can always find a real number arbitrarily close to an integer. If your metric space is the set points you described, then yes - for each point in the first example you can find some open ball around it that contains no othet element of the space, while in the second example, you can't do that, since the irrationals are dense.
Note that a space can have both isolated and non-isolated points. Take for instance the metric space $(0,1) \cup \{2\}$ with the standard metric. This space has both isolated and non-isolated points.
A: I think these are examples of countable and uncountable sets. If you want to define isolated points, you need a metric and the sense of a nbhd. However, you did not define these on $\mathbb{Z}$ or $\mathbb{N}$.
