Example of discrete set

Question

Please I need examples of discrete set and non discrete set... am a little confused over this expression.

I am thinking of discrete as a finite set but found from another article that, its not necessarily a finite set.

If I can just have two examples, I will be fine..

Thanks

Answer 1

The integers.

{$1/n| n \in \mathbb Z$}

Actually, any countable set of real numbers that can be put in order so that $.... a_i < a_{i+1} < a_{i+2} < ....$ will be discrete.

Discrete set means every point, x, has an open neighborhood in which x is the only point in the set. For any orderable countable set of real numbers indexed so that ... $a_i < a_{i+1} <$ .... then one can find a small neighborhood around each $a_i$ that only contains $a_i$ and no other point in the set.

Examples of non-discreet sets would be sets were at least one point is such that every neighborhood contains other points int the set.

For example. The Rational numbers aren't discrete because every interval around a rational will contain and infinite number of other rationals. The reals aren't discrete either.

The set {$1/n| n \in \mathbb Z\} \cup \{0\}$ is not discreet because for every interval $(-h, h)$ around 0, will contain points $1/n \ne 0$.